That's odd.
I've just double-checked (with a fresh Showkase v1.7.6 installation and the patch files from my box.com link) and I uploaded a .webp image to the Showkase Library as a test without any issues. (The image uploaded via the upload module without any problems or error messages and, afterwards, it appeared on the Images tab as expected.)
It looks like the files are fine.
The three most likely things I can think of that might be causing your problem (and, therefore, things you could investigate) are:
(1) The patch files have been uploaded to the wrong location on your server.
(2) When uploading the patch files to your web server, your FTP program has not been set to overwrite existing files.
(3) Your web hosting account has server-side caching enabled and your web server is still using older, cached files rather than the new patch files you've recently uploaded. It might take some time for the new files to become active on your web server.
Check that you're uploading the patch files to the correct location on your web server. If, for example, you installed Showkase to your root directory, then you would have uploaded the contents of the 'showkase' folder (from the Showkase zip file) to your root directory. If this was the case, then you would upload the contents of the 'showkase' folder (from the patch zip file) to your root directory.
Also, make sure that your FTP program is set to overwrite existing files (so that the patch files replace their original versions).
If you integrated the patch into a Showkase folder on your computer before uploading, make sure that your copy action replaced (and did not skip) existing files.
Finally, make sure that your web hosting account does not have server-side caching enabled. (Check in with your web host. They should be able to help you with this.)
I'll keep my fingers crossed that one of these suggestions helps you out.
Please let me know how you get on. Thanks.