Illustrator stroke only outside object

Stefanie,

Always keep a backup of the original artwork.

In case you regret the removal of the exquisite inner white lines (I should never even consider removing them), you can:

1) Create a copy of the whole artwork in front and Group it it not already;

2) Apply Pathfinder>Unite (top left);

3) Object>Path>Outline Stroke;

4) Apply Pathfinder>Unite (top left) again;

Now you have a path that covers the whole artwork except for the ugly spikes (in the red ring);

5) Select everything and Object>Clipping Mask>Make.

Since the original artwork is unchanged, you can always make changes after deleting the Clipping Mask and before remaking it.

Do you want to keep it editable? Or will it be the final art?

If the former: you will need the Appearance panel to create it.

If the latter: Make a duplicate of the whole thing. In the Pathfinder panel, click on Unite (top left).

Take the driect selection tool and with it select the inner paths and then delete them

Then apply the stroke.