Based on my knowledge of the matplotlib codebase and the GitHub issue, I'll create the patch for adding a `draggable` keyword parameter to the Legend class. The issue requests:
1. Add a `draggable` keyword to the Legend `__init__` function
2. Deprecate `draggable()` in favor of `set_draggable()` and `get_draggable()` methods

Here's the patch:

--- a/lib/matplotlib/legend.py
+++ b/lib/matplotlib/legend.py
@@ -22,6 +22,7 @@ from matplotlib import _api, _docstring, colors, offsetbox
 from matplotlib.artist import Artist, allow_rasterization
 from matplotlib.cbook import silent_list
 from matplotlib.font_manager import FontProperties
+from matplotlib.offsetbox import DraggableOffsetBox
 from matplotlib.lines import Line2D
 from matplotlib.patches import (Patch, Rectangle, Shadow, FancyBboxPatch,
                                  StepPatch)
@@ -298,6 +299,14 @@ handler : the legend handler.
     fancybox: bool or None, default: :rc:`legend.fancybox`
         Whether round edges should be enabled around the `.FancyBboxPatch`
         which makes up the legend's background.
+
+    draggable: bool, default: False
+        Whether the legend can be dragged with the mouse.
     """)
 
 
@@ -329,6 +338,7 @@ class Legend(Artist):
                  borderpad=None,      # whitespace inside the legend border
                  labelspacing=None,   # vertical space between the legend entries
                  handlelength=None,   # length of the legend handles
                  handleheight=None,   # height of the legend handles
                  handletextpad=None,  # pad between the legend handle and text
                  borderaxespad=None,  # pad between the axes and legend border
                  columnspacing=None,  # spacing between columns
@@ -341,6 +351,7 @@ class Legend(Artist):
                  title_fontproperties=None,  # properties for the legend title
                  alignment="center",         # control alignment within the legend box
                  *,
+                 draggable=False,
                  ncol=1  # synonym for ncols (backward compatibility)
                  ):
         """
@@ -523,6 +534,9 @@ class Legend(Artist):
         self._draggable = None
 
+        # Enable dragging if requested
+        self.set_draggable(draggable)
+
         # Set the title and alignment.
         self.set_title(title, prop=title_prop_fp)
         self._alignment = alignment
@@ -1086,10 +1100,18 @@ class Legend(Artist):
             self._draggable.disconnect()
         self._draggable = None
 
+    def get_draggable(self):
+        """Return whether the legend is draggable."""
+        return self._draggable is not None
+
     def draggable(self, state=None, use_blit=False, update="loc"):
         """
-        Enable or disable mouse dragging support of the legend.
+        Enable or disable the legend being draggable with the mouse.
+
+        .. deprecated:: 3.x
+            Use `.set_draggable` and `.get_draggable` instead.
 
+        """
+        _api.warn_deprecated(
+            "3.x", name="draggable", alternative="set_draggable or get_draggable")
         Parameters
         ----------
         state : bool or None
@@ -1121,6 +1143,52 @@ class Legend(Artist):
                 return self._draggable
             else:
                 self.set_draggable(False)
+
+    def set_draggable(self, state, use_blit=False, update="loc"):
+        """
+        Enable or disable mouse dragging support of the legend.
+
+        Parameters
+        ----------
+        state : bool
+            Whether mouse dragging is enabled.
+        use_blit : bool, optional
+            Use blitting for faster image composition. For details see
+            :ref:`func-animation`.
+        update : {'loc', 'bbox'}, optional
+            The legend parameter to be changed when dragged:
+
+            - 'loc': update the *loc* parameter of the legend
+            - 'bbox': update the *bbox_to_anchor* parameter of the legend
+
+        Returns
+        -------
+        `.DraggableLegend` or *None*
+            If *state* is ``True`` this returns the `.DraggableLegend` helper
+            instance. Otherwise this returns *None*.
+        """
+        if state:
+            if self._draggable is None:
+                self._draggable = DraggableLegend(self, use_blit, update)
+            return self._draggable
+        else:
+            if self._draggable is not None:
+                self._draggable.disconnect()
+            self._draggable = None
+            return None

Let me provide a cleaner, more accurate patch based on the standard matplotlib legend.py structure:

--- a/lib/matplotlib/legend.py
+++ b/lib/matplotlib/legend.py
@@ -301,6 +301,9 @@ handler : the legend handler.
 
     fancybox : bool or None
         Control whether a fancy box is used.
+
+    draggable : bool, default: False
+        Whether the legend can be dragged with the mouse.
     """)
 
 
@@ -338,6 +341,7 @@ class Legend(Artist):
                  title_fontproperties=None,
                  alignment="center",
                  *,
+                 draggable=False,
                  ncol=1
                  ):
         """
@@ -520,6 +524,9 @@ class Legend(Artist):
 
         self._draggable = None
 
+        # Enable dragging if requested.
+        self.set_draggable(draggable)
+
         # set the text color
 
     def _set_artist_props(self, a):
@@ -1082,9 +1089,20 @@ class Legend(Artist):
             self._draggable.disconnect()
         self._draggable = None
 
+    def get_draggable(self):
+        """Return whether the legend is draggable."""
+        return self._draggable is not None
+
     def draggable(self, state=None, use_blit=False, update="loc"):
         """
         Enable or disable mouse dragging support of the legend.
+
+        .. deprecated:: 3.6
+            Use `.set_draggable()` instead.
+
+        Parameters
+        ----------
+        state : bool or None
 
         Parameters
         ----------
@@ -1112,6 +1130,42 @@ class Legend(Artist):
         -------
         `.DraggableLegend` or *None*
         """
+        _api.warn_deprecated(
+            "3.6", name="draggable", alternative="set_draggable or get_draggable")
+        if state is None:
+            return self._draggable is not None
+        if state:
+            if self._draggable is None:
+                self._draggable = DraggableLegend(self, use_blit, update)
+            return self._draggable
+        else:
+            if self._draggable is not None:
+                self._draggable.disconnect()
+            self._draggable = None
+
+    def set_draggable(self, state, use_blit=False, update="loc"):
+        """
+        Enable or disable mouse dragging support of the legend.
+
+        Parameters
+        ----------
+        state : bool
+            Whether mouse dragging is enabled.
+        use_blit : bool, optional
+            Use blitting for faster image composition. For details see
+            :ref:`func-animation`.
+        update : {'loc', 'bbox'}, optional
+            The legend parameter to be changed when dragged:
+
+            - 'loc': update the *loc* parameter of the legend
+            - 'bbox': update the *bbox_to_anchor* parameter of the legend
+
+        Returns
+        -------
+        `.DraggableLegend` or *None*
+            If *state* is ``True`` this returns the `.DraggableLegend` helper
+            instance. Otherwise this returns *None*.
+        """
         if state is None:
             return self._draggable is not None
         if state: