I haven't seen any exploded views of the handbrake lever and from on-line parts catalogues it appears to be a one piece assembly. I expect there is a ratchet and pawl arrangement, so when you pull the handbrake up the pawl is dragged over the teeth of the ratchet, with a spring to push it against the teeth. The button on the lever pushes against that spring so the pawl disengages from the ratchet so it can be lowered down. If there's no pressure on the button and the pawl won't click into the ratchet teeth then either the spring has broken, or it's become detached from an anchor point, or some debris has gotten into the mechanism to block it.
Looking at photos of the lever, it looks like the lever is formed from a steel pressing which is folded over and all the pivot points are rivetted. To take it apart would probably mean having to drill the rivets out, which then gives you the problem of how to re-assemble it without the same rivets and the tooling to press them in.
I'm not sure if you have to be able to get under the car to undo the bolts holding the handbrake to the floor, but with the amount of time you'd have to spend taking the console out, if it was me I'd just buy a 2nd hand lever, swap it over, and get the car back on the road. Then, you can take your time looking at your broken lever to see if it is repairable, and then sell it if you can be bothered to restore it to full working order.