I'm sorry you feel that way Katherine. Not getting the answers you need can be frustrating.
To get back to your question, yes it is not the best thing to have a fish in a non-cycled tank. I absolutely do not advocate cycling a tank with fish because I think it is a lot of unnecessary stress. But, you were given the tank and a fish already, so just make the best of it! If you don't have an ammonia test kit, I'd recommend getting one so you can monitor your levels and make sure they aren't getting toxic. There's a product called
Cycle that introduces beneficial bacteria to your setup to reduce fish stress during the cycling process; I'd get some of that too. Plants help absorb nitrogen-based compounds like nitrate, so adding plants during cycling is good too. And I'd just get a new filter; it would probably be easier. I think the AquaClear (my favorite brand of HOB filters) for your size tank is about $15 and well worth the money.
In short, I'd go to the store, get an ammonia test kit, some test strips so you can monitor nitrite and nitrate (or you could bring water samples to your LFS, they usually do free water testing), a new filter (AquaClear if you can), Cycle, and some gravel substrate if you don't have any yet. I'll send you some plants today (java moss and hornwort which are both great for cycling). Ask your fish store if they'd give you some used filter media that you could seed your tank with; you can put their media in your filter and that helps beef up the bilological filtration.
Please ask as many questions as you like and I'll do my best to answer them. I'm glad to hear you're doing everything you can to keep this fish alive.