Hello - welcome! As you can see, there is a wealth of information here. If you have a question, just ask ~ usually someone has an answer.
I'm with Ann, most people do not have a CT scan when diagnosed with ITP. Sometimes people have an ultrasound of the spleen, but even that is not done very often. It's not on the protocol for diagnosing ITP. It couldn't hurt though I guess.
It's kind of soon to consider splenectomy. Sometimes a virus can cause ITP, or a medication or herbal supplement. Some people do go into remission within the first year, so having surgery is sort of drastic at this point. Also, as Ann said, the success rate is not that great after the age of 40. That can also open up a whole new set of problems to deal with, so you want to weigh the pros and cons carefully.
Sometimes the goal with ITP is not to achieve normal counts, but to be able to maintian a safe count. At your Mom's age, 50 should be a safe range. You don't have to panic if she falls below normal as long as she is safe.
You have quite a few options to consider. Be sure your doctor presents several of them. If he doesn't, I'd look for another opinion.