It is not uncommon for a computer located upstairs to not connect to the wireless router downstairs, depending on the router that you have. You don't really want another router, what you want is a range extender. This is a device that will take your wi-fi signal, amplify it back to 100%, and send it out again. This device will need a power connector but will not need an Ethernet cable. The best place for this device is directly in the middle between your router and your desktop computer. That way it receives a good single from the router, but cuts the distance in half to the desktop computer. It just needs power and is very easy to setup.
If you wanted to test the idea that your desktop is indeed to far from your router to receive a wireless signal consistently, then move your desktop downstairs and see if it works fine. If it does, you know it is the distance. If it does not, I would start by replacing your dongle. Taking your laptop upstairs to see if it receives a consistent wireless connection from the router is something you could try, but the laptop probably has a better wi-fi card then your dongle, and so it is not a super efficient test. It would be worth a try though before moving a desktop computer. The whole problem may be in your dongle.
You can purchase a wi-fi extender here:
They are not very expensive and I certainly wouldn't go to a lot of trouble doing anything else, just grab one of those. That is probably your problem.
As far as which one, Newegg has a lot of customer reviews. Only other thing I would mention is I would get one that is the same brand as your Router....it shouldn't matter, but they might play together a little better.
Setting up two routers in a setup like you mentioned can be done but is not easy. I wouldn't even try it, as you need the right routers and a lot of know how to get them setup. Just get the extender and you should be good to go, it'll take you 10 minutes to setup and you will be all set.
Hope this helps