It's all about the focal length numbers.
Kit lenses are generally 18-55mm. 18mm is the wide end of the lens. If you want something wider than the kit lens you'll need something with a shorter focal length. IE the Nikon 12-24mm.
But if you want something cheaper than that Nikon, try looking at a Sigma 10-20mm lens, it will be much cheaper (prob £3-400) and still good quality.
However, the 18mm end of your kit lens is probably good enough for most photography situations.
Your Nikon D3100 is a good enough camera, sounds like you could do with investing in some better quality lenses first before upgrading the camera.
Once you've sold the Sigma, look at buying something like a nice prime (fixed length) lens, like a 30mm or 50mm with a nice wide aperture (f1.8 or f2.

which will give you some quality glass to take photos with.