DVI is compatible with HDMI, an adapter plug from one to the other costs about $1.
If you’ve only got VGA then you’ll need an S-Video to VGA adapter for $13:
That should work with your existing HDTV converter boxes. It has a VGA input, perhaps that will avoid the need for a VGA AB switch / (aka KVM)? Not sure.
If you did have a DVI input in that monitor, the HDTV picture could look much better.
NOTE: You will need to wire up the converter box to your computer speakers as well.
2nd NOTE: The VGA adapter gets power from USB, old PCs cut power to USB ports when they’re powered off, newer PCs tend not to. Might end up needing to use a phone charger or similar to power it.