Here’s a link to the mentioned userscript Anki Mode. There you’ll find further information how to install it.
Just one little tip after you successfully installed it and want to use it for your next reviews: Since WaniKani changed the behavior of their site, you have to refresh your review page (e.g. via hitting the key F5 on your keyboard or clicking that refresh button on your browser in the top next to the address bar) so the userscript actually starts working. (If I understood correctly, just by switching from your dashboard into the reviews via the review button, the page doesn’t actually reload/refresh in that sense, so usersripts can’t recognize that they have to do something.)
Oh, and just keep in mind: userscripts are a third-party-script from WaniKanis point of view. So whenever they change something on their page/server, it might break the script and one has to wait until the creator updates their script. (But you can always deactivate scripts, so you still can do the reviews as before - well, just without the deactivated script, of course)