How about trying for a 5k first? Starting out, the training for a half is going to be about the same as for a 5k anyway, and it's a more reachable goal even if not as exotic.
You can train without running more than 10min at a time. Just take a 2-minute walk in between runs. And run slowly. No deep, rhythmic breathing.
I have considered maybe a 5k or 10k as an intermediate distance before the full half. It's true, a half marathon has the allure of being exotic-- I think that's why I have the motivation to do it, though. My PT says he can retrain my running form in only a couple of weeks, so I'll check back in as he tells me what's feasible and what's not.