Due to the latitude of its launch site and overflight issues with China, the minimum inclination you can launch into from Balkinor (Tyuratam) is about 51 degrees. That's why the ISS is in a 51 degree inclination. Columbia was in a 39 degree inclination. Neither a Soyuz or Progress could've performed enough of a plane change to rendevous with Columbia.

As for launching another Shuttle, it might've been possible had they known of the problem on the first day. However, what if the 2nd orbiter experienced the same problem (after all, they still don't know the definitive cause)? You could end up with two orbiters stranded in space.

I don't know how hard it would've been to quickly send up an unmanned cargo pod into Columbia's plane (letting Columbia handle the rendevous maneuvers). There may still have been other problems due to Shuttle limitations, but it might've worked had they known soon enough.