Macbook Pro doesn't switch on unless I take off its battery (mid 2010)

Li-ion batteries do not respond well to being allowed to run absolutely flat. If they drop below approx 2% power, a firmware switch will kick in & not allow then to be recharged.
A professional may be able to circumvent this, but it is potentially hazardous & should not be attempted by the user.

Your choices are…

  1. Leave it on charge for another day & see if it's actually charging, albeit slowly.

  2. Take it to a repair shop & see if they can 'reboot' the battery.

  3. If it will run without the battery, then try that. It will throttle down & will probably crash hard if you stress the CPU or GPU, as the charger alone is not powerful enough to run the computer under high load - that's what the battery is for.

  4. ..of course… buy a new battery.