I have and maintain a number of older Laptops for family and friends and over the years replaced quite a few batteries. A new battery will not improve the speed, you need to ensure that unnecessary files and software are not causing it to be sluggish. That can be done while running directly from the power adapter before you decide whether or not to invest in a battery.
If you do decide to buy a replacement battery, it is worth checking the returns policy before ordering - choose one of the reputable suppliers, that will cover the return carriage if it turns out to be faulty or incompatible. Some Laptops are fussy and will not recognise 'so called' compatible replacements, so simply won't charge them. The cost of genuine batteries may make you think twice about spending so much on an older Laptop.
Pete
In 3 days, a completely turned off laptop discharges by 6%