There are a couple of things you could do....
Option 1: VGA to HDMI Adapter
The cheapest one I found with relatively good ratings was (at the time of this post) $17 available on Amazon. VGA being an analog signal and HDMI being digital, there's no getting around the fact that you have to convert the signal.
Note: You will need to supply your own HDMI cable to go from the adapter to your TV
Option 2: Upgrade Video Card to DVI and use a DVI to HDMI Cable
You said you had a desktop computer, so you could easily add (or replace) your existing video card with one that had a DVI connector. These are selling for as low as $5.95 (at the time of this post) on eBay. Keep in mind that this is going to be a used video adapter, but given the price constraints, I don't think this will be too much of an issue.
You can then use a DVI to HDMI cable to connect that video card to your TV. Amazon has it for $6.95 (at the time of this post). Since DVI and HDMI are both digital, no digital to analog conversion is necessary
Unfortunately, there are no options that I could readily find for under the $10 limit. The second option I proposed seems to be the cheapest, coming in at about $13 USD.