Who is Ichigo's father really?

I think it's all coming together now. Isshin used to be a captain. He was a captain during the time the Quincies were being exterminated. It's possible that a bit before the extermination campaign, Isshin was on a mission and met Masaki.

When the orders were given that the extermination would take place, he made plans to escape with Masaki to the human world. I'm thinking that the head captain or someone of high power found out about these plans and tried to kill Masaki and him. At this point he could have used Final Getsuga Tenshou, effectively making him lose his powers indefinitely.

I think the above is more plausible since during the Pendulum arc, we see no signs of Isshin, so he probably escaped to the Human World before the whole Hollowfication experiments took place. (causing Urahara and Yoruichi to escape using Gigai)

Once Urahara and Yoruichi escaped to the human world, it's very possible that during this time they helped Isshin regain his powers, or at the very least, allowed him to secretly train in Urahara's secret underground training facility.

Isshin being a family man actually supports the fact that his rate of training, and thus reawakening his powers was severely limited. Even though Ichigo comes and goes as he pleases, he still has to protect his two daughters when Ichigo isn't around.

Ichigo regained his powers so fast because he was stimulated to do so, plain and simple, like someone else mentioned.