Does your slimline have wireless that is broken, or did it not come with wireless to begin with? We'll need the following information to help you:
Is your broken wireless adapter an internal card (mounted inside the computer case)?
If internal, what interface does it use: PCI, PCI-express, or other?
What kind of wireless signal do you need? Wireless g, wireless n? For wireless n, is your router a dual band rounter (2.4/5GHz)?
The quick fix for a person unfamiliar with the inner workings of a computer is to purchase a USB wireless device and just plug it into an available USB port on the back of your computer. Here is a relatively inexpensive USB wireless adapter that can replace your broken wireless or add wireless if you don't have it to begin with (link) and it can work with wireless b, g, and n.