Congrats on seeing the light and choosing the infantry!
Seriously, with the CF being funded the way it is, my unit is seeing a steady trickle of people who have realized the only way they are going to get to do real "army s**t" in the reserves is by joining the infantry. It costs money for all training, but light infantry training can cheaper to conduct than training for other trades.
This is the best way to get your transfer started. ( I know that you put in a letter already, but if you submit a proper memorandum than your chain is obligated to respond to it.)
Put a memo in to your section commander stating that you wish to remuster. Be specific as to what you want, (Unit desired, reason why, etc.), and it never hurts to request - politely - that you would like the transfer to be completed by a reasonable time. Check with the unit you are transferring to by calling their Orderly Room; this is a good way to find out available dates for a BIQ course - which is the date you want to be in your new unit by. Also by making yourself known to the unit are are transferring to may also prove to be beneficial in expediting this process.
Keep your transfer in the front of your chain's mind by asking about it at regular intervals. Continue to train with your unit. DO NOT allow yourself to go NES or give your superiors any reason to discipline you. Keep copies of all your memos that you submit regarding your transfer as you may need to refer to them later on . . . especially if your transfer gets stalled somewhere along the line.
As to how long this will take; I wouldn't want to give you an estimate as I have seen time spans right across the board on this issue. If it takes any longer than a month to get some progress back from your current unit, however, I would start to ask questions.
All that being said, it can be a fairly simple process once you get it going along. Best of luck to you.