I think the first think you need to do is get the balance wheel with attached hairspring out of the way. You don't want to damage that any further
Put a piece of rodico under that fine regulator arm to hold it in place while you remove and turn over the balance c ock.
Then you need to unscrew the stud retaining screw that holds the hairspring stud in the balance c ock. That's the screw at the 7 o'clock position in your photo.
only turn that screw half a turn at a time, until the stud is released. You do not want that falling out!
Where the terminal coil of the hairspring passes through the regulating pins, the 2 brass pins at the 8 o'clock position there is usually what's called a 'boot' that sits across the 2 pin to prevent the hairspring from jumping out of the pins.
This 'boot' will rotate. They sometimes have a tiny slot in the top to get a screwdriver in, they sometimes have a hole in the side to insert a tiny needle etc. or grip it with a pair of tweezers and turn it a quarter turn and lift out the hairspring.
Remember to do the stud retaining screw up as soon as you have the stud out.
I can't help you with the refitting the regulator arm as I haven't done one yet.
But get the balance and hairspring clear first
Good luck.