first things first - integrated sound cards /aka the one that is on your motherboard/ really suck. so your best option is to get a decent discrete one /which fit into the pci or pcie slots like your gpu fe - xonar dg or dgx cost about $30 and is a huge improvement, sound blaster z or titanium hd cost way more but are currently the best for games/.
second - 5.1 or 7.1 doesn't matter much if you haven't got enough channels /aka small speakers in each cup/. and considering your budget there is no way your headphones will have more than 1 per ear.
so, the only way to get surround sound is to cheat a bit /see the barber shop
https/youtu.be/IUDTlvagjJA and it's use of virtual surround/. that is what any decent sound card can do and that is what lots of usb headphones use /and have an integrated chip for that, what is the reason they are mostly utter overpriced pieces of garbage/
btw.. I have got hyperx for my mother and she doesn't nag, so I guess it is comfortable enough and the sound is clear enough /at least for skype, etc.../