How come the MPs didn't notice Annie was gone all that time?

Once you start pointing out plot holes in anime you might as well stop watching it as it is full of holes. I mean, this show has giants, people flying using wires, teenagers taking lead in a strategic military operations... etc.

Come on, man.

I think there was a visual novel that gave some details on that whole thing (Annie leaving to attack the 57th Expedition of the Survey Corps), and I have been able to find about all of nothing on it, including on the AoT wiki. It's called either Goodbye Wall Sina or Wall Sina, Goodbye.

"Greetings! This is not God, but his close friend, Officer Boscorelli. Please pull over."- Bosco

They're corrupt or incompetent. They just don't care.

Didn't they think she was sick? I recall someone making some comment like that when they first show her after all the female titan shenanigans.

You'd feel cocky too if you were full of myself.