Oracle's deployjava.js is no longer functional in Firefox/Chrome. What else can I do?

Question

Since Firefox and Chrome blocked Java Plugin, Oracle's deployjava.js used a trick to detect JRE - by using navigator.mimeTypes array and locating Java's mime types inside.

But Firefox no longer exposes navigator.mimeTypes. Since version 53 it is an empty array.

For Chrome, the array is still accessible but no longer contains Java's mime types.

I already migrated my applets to using JNLP, but I still need to make sure the client has a suitable JRE, otherwise the end users will be looking at a dialog asking them to pick an application to launch the .jnlp file, and they won't know what to do.

What other methods can I use to verify JRE on client computer from my website?