You can really use both without any problem. The "real" inflected Konjunktiv II form in your first variant
Wenn ich ein Kätzchen bekäme, würde ich mich sehr gut um es kümmern.
is on a higher level of language, and you would probably slightly prefer it in written language, especially in your example, where it also avoids the double "würde".
The main clause in your sentence is also a good example of a case where the form with "würde" is always preferable, namely when the inflected Konjunktiv II form is indistinguishable from the Präteritum form:
[...], kümmerte ich mich sehr gut darum.
This would be unusual nowadays, the "würde" form is much more common.