It may be worth reconsidering your deduced zero force members. But for this question it is not really necessary to determine which are zero force members in advance as this will come out in the analysis.
It is not important if the truss is determinate or not in order to get the member forces your are interested in (highlighted in red). Approach this question using the method of sections. I have marked sensible section cuts in green. You can get the member forces of JL and JK by considering the equilibrium of joint J. It is worth taking note that the member forces requested hints at the appropriate solution method.
For example, in order to get the member forces RT, RS, and QS make cut 3:
Notice that since you have the reaction forces at A already the member forces RT, RS, QS can be solved by equilibrium.
Getting member forces for NP, MP, and MO is a little bit more tricky, but once you have member force for MH (by making cut 4) you can make the cut marked in blue.
It is a bit tedious using method of sections, but there isn't a faster method that I am aware of. Even modelling this in a structural analysis software package would probably take longer ...