It may be a little bit tricky, but the main thing to know is that DVI has the same signal as HDMI so it should work
The problem is sound
But you have to tell us more facts.
The minimum is exactly which Mint you have - which version and which edition (main or any of the CEs)
We also need to know a bit about all your hardware.
Run " lspci " in a terminal and post here (to avoid misunderstanding - the first character is a minor L and you copy by marking with the mouse and then right click)
Some wifi and possibly built in sound can be connected to USB even if they are internal - if you suspect you have one of these run " lsusb " in a terminal.
Also tell us if this is using mint4win or not, quite important
Don't fix it if it ain't broken, don't break it if you can't fix it