czy to zdanie jest dobrze??

First thing which had to be done was finding out who the woman from the car was, so detective Monk asked police officer to check all the disappearances reported recently.

seems all right but maybe wait for the opinion of someone more qualified than me ;)

imho
First thing THAT had to be done was TO FIND out who the woman from the car was, so detective Monk asked A/THE police officer to check all the disappearances reported recently.