Does the Wifi adapter appear in Device Manager after installing the chipset drivers? Are there any yellow exclamation marks next to any devices in Device Manager.
If after installing the chipset drivers no Wifi adapter is present in Device Manager it's possible the wifi card has failed. The chipset is a Realtek RTL8188EE according to this page. The chipset is not common and is not on Realtek's home page. It's looks like a chipset specifically made for OEM's, HP being one of the OEM's. You may want to pull the card and attach it again. Pull the power, hold the power button down for 30 seconds. Ground yourself on some bare metal of the case before attempting to pull the card.
You can either check the card by booting a live linux disk like Mint or purchase a USB Wifi dongle. Because it's a desktop you can purchase a dongle with antenna for better coverage.