Often when I come to visit my mom (97) she is sleeping deeply. Should I keep trying to talk to her or let her sleep?

I would wake her up. She can go back to sleep if she wants. I know my mother would want to visit family and would be upset if she missed them. If she falls back asleep sit quietly next to her, hold her hand and perhaps sing a soothing song, play some soft music and just be there for her.

If she's sleeping, you could try reading out loud to her. My mom likes me to read the bible to her. Even if she's asleep she might still be comforted by it.

What do caregivers tell you about the best hours to visit? Is there an hour when they awaken her to assist her with meals? Is there an hour when she is bathed and may be more awake? Is there an hour where they get her up in a chair.
There is no reason not to awaken her when you visit. She will drift off to sleep again when you leave. Unless this is agitating or upsetting to her. Your goal now is her comfort and happiness and well being. Whatever works best toward that goal is what you should stick with doing. Try it several different ways and speak to her caregivers to some ideas of what might work. Ask them if she asks for you when awake.

Ask the staff when she is more alert. If it is at meal time try visiting then.
If this is not possible when you visit if she is asleep leave a note for her and if you bring something special for her each time she will see that. Some flowers, a cookie, a favorite piece of candy. She will know the gift is from you. A white board would be nice for the note or a bright piece of paper that will catch her eye.

I found it helpful to visit around meal times because I knew that my mom would be awake then. At other times if she was sleeping her eyes might flutter when I said hello but she usually just sank back into her doze so I would just hold her hand as she slept, I usually took a book along so I could read a chapter before leaving.