One thing you did not mention trying is a CMOS/BIOS clear, which afterwards on a start up should display a screen giving you the option to enter the BIOS via the Del or F2 keys. That screen has a timer which counts down (~10 seconds) and if no keys are pressed the PC will boot with all default BIOS settings. But the BIOS clear ought to allow you to get into the BIOS.
Your board seems to only have the jumper to reset the BIOS, some ASRock boards have a button on the IO panel. So you'll need to open up the PC, find and move the BIOS clear jumper. That procedure has not changed since your last PC build.
Actually your issue might only be a problem with the video card, given no output to the monitor. As long as a video card is installed, the BIOS will select it as the video source. Connecting a cable to the on-CPU video in that situation will do nothing, you would need to manually select the on-CPU video in that case to override the BIOS selection and current setting of the card as the video source. IMO, you should remove the video card while you have the PC case open to do the BIOS clear, and just use the on-CPU video.
Are you getting any video output to a monitor now? It seems like you might not be. I'm not sure you are actually having an issue getting into the BIOS, you just can't see what is happening apparently.
I've seen my ASRock Z97 board have keyboard recognition issues at times, which might explain no reaction from pressing Del and F2. Use USB 2.0 ports for the keyboard and mouse, it seems the USB 3.0 ports are not awake during POST, but that might be due to my PC's configuration.
The inability to enter the BIOS UI you mentioned happens with a UEFI booting Windows 8 installation, and having the Fast Boot option in the BIOS set to Ultra Fast. I you aren't familiar with this, then you didn't do it.