There doesn't seem to be any easy fix that I can find, nor any fix that works for everyone. Steam sells a download, but only lists Windows 98/ME/2000/XP and not newer.
A couple of people in this Steam thread
https://steamcommunity.com/app/11130/discussions/0/490124466477849812/?ctp=2said they got it working in Windows 10, but that was two years ago -- and Windows 10 two years ago is a different OS from Windows 10 now, so suggestions may not work. Some people in that Steam thread mention tweaking the DxWnd launcher that is included in the Steam version, and one person wrote a how-to for using it here
http://steamcommunity.com/sharedfiles/filedetails/?id=1310996482 Unfortunately some people say the DxWnd launcher made things worse.
pcgamingwiki only mentions a fix for crashing cut scenes.
https://pcgamingwiki.com/wiki/Sherlock_Holmes:_Mystery_of_the_MummyFrogwares support for the page doesn't help for mouse
http://mail.frogwares.com/support/kb_cat.php?s=beee64bffe1a9f5342f34ebd30211bcf&id=3&t=qandaIt mentions a Steam page about DxWnd for fixing a flickering screen problem.
http://steamcommunity.com/app/11130/discussions/0/846958724802436682/#c648811126470323751Unless you want to try fiddling with DxWnd settings, I don't think there's much hope for getting it working on Windows 10. It's hosted at sourceforge.
https://sourceforge.net/projects/dxwnd/Best way to play would be on an old computer with Windows 98, ME, 2000, or XP installed (if you happen to have one).