It also needs to get as close as possible to the top of the hill without going over. L2 is not a stable point: you’re either falling back to Earth or drifting away. If it goes over it can never get back as it can only fire in one direction - away from Earth (to protect the instruments from the Sun). So it’s a continuous balancing act where it falls downhill towards Earth a bit, then jumps up without going over the edge, rinse and repeat.
This answers a question I’ve had for a long time: how can the telescope station-keep without any thrusters pointing “up”, away from the sun? Answer: very, very carefully.
> How long will it be breaking? (If at all, or of it is not already doing it).
All* of it's acceleration was given by the rocket at launch, it has been constantly "braking" since then as it's being pulled by earth's gravity, and it will reach L2 at same time it reaches a speed of 0.
* Not actually all in truth, since for margin of safety reason, the Ariane rocket purposefully imparted a lower than necessary speed, and the more precise thrusters on the JWST will be used to top it off with 3 different short burns (2 of which already happened), always staying just short of the required speed. The idea being that you want to make sure to be always missing a little bit of speed and readjust as needed, but never too much, as that is not recoverable, and the telescope would just drift away for ever.
Right now, it is moving 0.3933km/s which according to Google is 1415.88km/h. That is pretty fast.
What speed does it need to be at for insertion and steady state? How long will it be breaking? (If at all, or of it is not already doing it).