You are slowly killing your HD. That sudden shutdown can mung up the MBD (master boot directory) so that you disk is unusable. So the slowdown is because the MBD and essential files on your machine are fragmented. Only stopping the crashes, reinstalling the OS on a reformatted (non-fragmented) HD can put the life back in your laptop.
The battery should power the CMOS (in the Mac world that's called the EFI) many Mac Laptops do not have separate CMOS (backup) battery - some do. The EFI/CMOS stores certain information bout your machine among them the date and time. That default is an easteregg to the programmer who wrote part of the OS. It also lets a user know he's killed all the juice to the machine.
Every time you crash the OS you risk corrupting essential data files you can get away with it once or twice (a few times acutally - like running a stop sign, not everyone is killed the first time the run an intersection, but the odds are not in your favor). But if you continue in this fashion your machine will go from slow to frozen... in non-startup mode.
Apple states that running late model laptops without a battery slows them down.
You need to replace your battery ASAP and SWITCH OFF THE COMPUTER before unplugging it.
If this answer is acceptable please remember to return and mark it.