Just a thought Jooshy how about making a bootable Ubuntu disk > load the no install option > copy the driver while in the Ubuntu OS to a stick.
Reboot to Windows ands load the driver from the stick? Got to worth a shot mate.
I cannot think of why this is happening as I just Googled it and it is there but why it will not download. Realtek RTL8188CE 802.11b/g/n Wireless LAN Driver for Microsoft Windows | HP® Support
Can you tell us what is in the Device Manager - Network Adaptors > click on Network Adaptors find what is listed. If the drop down list has the adaptor listed has it got a yellow triangle alongside? If that is the case right click on it go to Update driver Software > click it and use the auto search option for the driver. Come to think of it try that anyway.