I use the 64 bit version of 11.0 and have installed the standard version of konqueror and java 32 bit version (plus plug-in). This works fine for most java based sites.
I have the 64 bit version of Firefox (which obviously won’t work with java) and as a back up for the odd java site that won’t work with konqueror - I have installed the 32 bit of seamonkey. This works perfectly for the dozen or so java sites that I have tried.
Using openjdk/icedtea will fix Firefox but afaik it will replace java, so you need to re-point konqueror to openjdk/icedtea.
I now have java on both FF also in checking at Sun Java it will tell me I’ve not got it. Yet this site never lies don’t know why but using the Sun site will not confirm icedtea is working and I also presume openjdk.
Konqueror uses Java directly or something, but firefox needs the plugin which isn’t available by Sun’s Java but Openjdk/icedtea have an implementation.