I accidentally ran into my open microwave door and I think I twisted it. Ever since then I have to slam the door (more than a few times) and push in the upper right part of the door by the latch for the light to go out and start. I do hear like a distinctive click that seems to be coming from the other side (hinge side) when it does get properly seated and start. It is a Kenmore model 665.69682992.
Buy a new microwave.
It sounds like your microwave door is pretty cattywampus. Microwaves are carefully engineered and undergo extensive testing to ensure radiation is kept in, and are typically not intended to be user serviceable. Even a very small leak could have serious health consequences over time. Finally, if you are able to replace a door, you still don't have the equipment to verify that there are no leaks. In short: it's not worth the risk!
Try placing a dishrag pad at the corner of the opening that makes contact first and giving the door a few gentle but firm shoves to reverse the twist. Use caution if there's a glass panel in the door.
If the door can't be restored to good working condition, replace the door or the entire unit. Microwave radiation isn't something to be taken lightly.