Not sure if this is the right place to post this but what the *censored*. So I've been given a desktop for free because it's broken, I was initially going to just take the hard drive and bin it but it seems to turn on fine (no error beeps on start up). The problem is I'm not great at this sort of thing and I don't really know what's wrong with it, so here's a run down of what's going on:

The monitor gets no signal from the tower (the monitor works)

The keyboard and mouse get nothing EITHER (no lights etc.)

I also can't turn the tower off with the power button

However the usb ports seem to work and as I said it turns on fine. I'm really tempted to just stick with the original plan and get rid, but if it's not that difficult to fix (or at least figure out what's going wrong) I'd rather do that and have a free desktop obviously. Any ideas what I can do next/what the issue might be?First thing I would try is to remove the power, remove the CMOS battery, let it sit for 30 minutes, replace the battery and try it again.  This will reset the CMOS settings.  Also, check the power connections to the motherboard, specifically the 20 or 24 pin main connector and the 4 or 8 pin supplementary power connector.
If that doesn't help, check how much RAM is in there, and if it has more than one stick, try them one at a time.
If it still does the same thing, check if it has a video card, if it does and it also has onboard graphics, try removing the video card.
Finally, unplug everything that isn't necessary for it to POST, for example front PANEL USB connectors, HDDs, and any other expansion cards, and see what happens.The other thing is since you inherited it even though it's a desktop they are sometimes not as rugged as you think...moving and transport who knows what could be jarred...
Remove a side panel and remove and re-seat the RAM, vid card if applicable and any other PCI cards in the system and re-seat them...
While in there do the housekeeping and a good dusting with a can of compressed air...
Re-assemble everything... powerup...and BEST of Luck.