I would suggest a single slot card that doesn't require additional power via the PCI-E 6/8pins. GTX1030/GTX750//GT730 come to mind that draw power off the PCI-E slot however they will need to be single slot.
You can follow this guide
on hot to disassembling the unit and this thread of someone trying to do the same thing
You could prove me wrong with dimensions and shoot a pick of the unit disassembled. Your warranty has expired, correct?