Can you have a DSLR refurbished?

I have had mine (a Nikon D5000) for 3or 4 years now, since it first came out. It was my first camera of this type and to be fair I haven't treated it as well as I should have. It still works ok, but the piece of rubber round the viewfinder is broken, and I'm just worried that the bits and bobs inside could do with being tightened up, or cleaned- can that be done? I also need a new lens cap.
If so how, where, how much?

Added (1). Oh and a couple of tiny screws on it look bit rusty now.

Yes, Nikon will clean and restore you camera.

I recently sent one in to have the rubberized grip reattached and when it was returned it looked like new, none of dust or corruption that had been on the camera was evident upon its arrival back to me.

Just ask for an estimate for the refreshing of you camera and its bits back to new before sending it in

Yes, send it to a Nikon repair centre; you'll find an address on their web site.
By the way, you can buy a new lens cap far cheaper from any camera store.

Please! Treat it with more respect afterwards… DSLRs are precision instruments and it's so sad to see them get tatty.

You can have it serviced. Ask for a maintenance check up and some CLA (clean, lubricate, adjust). Refurbishing means something else.