About two days ago I was unable to get into the BIOS setup and this is currently true now as well. It happened after I acquired a new monitor and keyboard as well. It did work once, however, but as I say is not now working.

Do I need to strip everything down to the bare essentials to find out what's wrong then?

What ever key you used with success in the past, try holding it down the whole time during early post- (Rather than "tapping" it).
Then, when signs of activity seem to slow or stop, let go.

You may get an error, but it's worked for me on many a stubborn Computer.

Have you tried hooking your old keyboard back up? If you went from a PS/2 keyboard to a USB one, your BIOS may not be set up to use it.