What transistor to use?
Yes, it will probably work although the KC637 is lower frequency than the 2N2222. This circuit probably operates at ~1MHz which is well below the 50MHz ft. The other parameters look better or okay enough.

The pins are in a different order compared to the 2N2222 so you will have to account for that. The 2N2222 is E-B-C and the KC637 is C-B-E. So, just flip the transistor over. Also both are different from the BC637.

Pay attention to the direction of winding on the coils.