When CIC requests additional information or documents, they normally place what's called a "bring forward date" on your file - so, although it doesn't go to the "bottom of the pile", it will be set aside in a pending queue awaiting your response and, for example if they allow 90 days for you to submit the information, they're going to assign a 90 day BF date and they won't look at your file to see whether the requested info has been received until after that date has passed. That's probably why your lawyer is not in a huge hurry to get info from you and back to them.
As far as the MP - you don't get a letter from the MP. The sponsor can simply contact the MP's office and ask them to make an inquiry on the status of the file. Some MP's offices are very interested/cooperative in doing this - others aren't. It all depends - but normally an inquiry from the MP's office will result in nothing more than a form letter from CIC that quotes the standard processing timelines. Your MP cannot do anything to "speed up" processing of your application. In fact, most of the time, even when there's a problem with processing, the MP's office will not be able to do much, if anything, to resolve the situation or even to find out what's happening.
You didn't say whether you filed inland or outland. If you filed inland, you shouldn't expect to hear anything for at least 5-6 months. And that's just to first stage approval. After that expect another 6-12 months to finalization. If you filed outland you can check the
processing timelines for your particular visa office. If your file falls within normal limits for that visa office, don't be sending periodic emails to the officer processing the file. It has a tendency to annoy them. Even inquiries from the MP's office as to the status of the file, if processing is still within normal timelines, will not be welcomed. If your case has been in process for awhile, and you're at the upper limit of the processing timeline, you can
order your CAIPS notes (FOSS if inland). This is a computer print-out on the status of the file that will sometimes include notes from the IO about any potential problems. Receipt of the CAIPS notes takes about 5-6 weeks.