First, I really do admire your efforts to do the very best for your students whatever you can. And fortunately, your report contains sufficient details to make a realistic assessment as far as this can be done from afar.

Normally, I would say that you should divide the classes according to less advanced and more advances students. But I conclude that you have actually have both groups of students in a single class for reasons beyond your control. It would beneficial for your students if you told them very clearly that the needs for those preparing for a college entrance exam essentially differ from those who wish to become more proficient in conversational English. Lessons for the former group are more formal and stringent in nature than those for the latter group, and also the topic to be covered essentially differ. At least, you could advise those preparing for a college entrance exam that they might need to extra classes somewhere else to have those special topics covered that you may not be able to cover. Please take this as a side note which I feel should be added.

Now, let me turn to your concrete question:

I just need to know if I should teach parts of speech and spelling rules from my old reading program.

From my point of view, the answer would be definitely YES - provided that your students can understand and apply them. It's not ever student's cup of tea, and some students respond differently to such a teaching approach than others. I suggest you give it a try and see how it works out for them. Giving your students rules for spelling and to teach them the parts of speech (grammar-related lesson parts) helps them grasp and perhaps also master the more intricate aspects of practical English language skills, both written and spoken, that are determined by such rules. This makes them less depend on the teacher and serves to strengthen learner autonomy. The latter is the best thing that a dedicated teacher can do for his/her students. After all, this is then basic role of a teacher - to assist his learners/students succeed!

Good luck in your efforts!