I think it's fine to do it that way for brief stretches, but I wouldn't do pages of it. You might also consider only writing out the highlights, and avoiding having too much chat speak by glossing over the fluff by saying things like, "We picked a restaurant for Friday night and set a time: 8:00pm."
I'd also consider how grammatically correct you want to make your chat speak. If it's written in leet speak, people may not want to go through it. I'd recommend keeping it pretty correct, with proper punctuation and capitalization &c. It's probably less realistic, depending on who's typing (although I always chat with correct spelling and grammar), but it'll be easier to read. Just assume they're very anal about proper grammar. It can work with more realistic spelling and grammar, but I think it'd be more likely to turn a reader off.