Sempervivum?

Skye

It is probably Sempervivum as Embothrium suggested - based on info you supplied. It is difficult to ID from photo, since it is in bad shape.

If it is Sempervivum, it should be outside whole year round - depending on where you are. They are hardy to z4 and some sp even z3.

I am not sure if it will recover; maybe yes, maybe not. Just wait and see.

I would suggest to keep it cool and dry for now. What are temps where this plant is?

It has only ever been inside since I got in in August and I live in Central Illinois. It is generally around 65 degrees here

It should be outside, especially in your location. It is 15F here right now (2pm) - and during winter as low as -10F and below. They 'sleep' during cold weather, just as most hardy perennials do. If you have outdoor space, put the pot outside. I would keep it in shade since it is in bad shape, and if it starts improving - move into more sun but gradually. If it is that warm there, I would water it but only when existing soil dries up.

If Sempervivums blooms, that rosette dies. You said it did bloom, do you have a photo of blooms by any chance? They should look like this or very similar:

Young 'extensions' could be chicks. Sempervivum should grow lots of them, to replace blooming plant (that dies when blooms are finished). They look like this or very similar (these in the photo are quite etiolated, that is how I bought the plant):

I think this has been slowly dying since it flowered. Normally it would have made chicks before flowering, but I suspect who ever gave you this plant took the chicks off. But, the plant is/was still trying to make new chicks. They are really small, but they might survive even if the main plant does not.