What I was looking for is the model of your computer, from that I can help you do much better. What i didnt notice ewas your link...
For the record, theres plenty of youtuber videos of people playing LoL on the internal video card.
From my newly discovered link, you have a 300W power supply.... Which is just squeeking by...
With only upgrading your video card, this:
Newegg.com - EVGA 01G-P3-1335-KR GeForce GT 430 (Fermi) 1GB 64-bit DDR3 PCI Express 2.0 x16 HDCP Ready Video Card
This isw your best option, its $50 roughly,
Your more affodable option is a gt 520, at anywhere from $20 after rebates and up
Newegg.com - XFX GT-520M-ZNF2 GeForce GT 520 (Fermi) 1GB 64-bit DDR3 PCI Express 2.0 x16 HDCP Ready Low Profile Ready Video Card
Your limiting factor is your power supply only being 300W. I would not recommend putting much money into your system, as its getting pretty dated, and a brand name.
I'd also recommend what ever card you get, get one with a fan, your HP CASE has poor airflow to use a "silent" card.