Mid 2010 MacBook Battery Replacement Recommendations
I got the same Macbook Mid-2010 which is valuable to recover to do a lot of stuff. I would recommend you to find the trusted vendor for the battery and other parts as we might have different area. See the review online to check whether they are selling good parts or not. If your decision fall to iFixit hardware, it could not go wrong as they are trusted and reliable as well with complete instruction for you macbook as well.

I would recommend you to learn to clean the macbook as well, buy thermal paste, change it yourself carefully which can be fun and add knowledge. In summary I would say:


Clean the macbook inside out and change thermal paste (buy thermal
paste if you do not have one)
Replace HDD to SSD in your macbook (buy HDD caddy to replace your optical drive and use current HDD as extended storage)
Upgrade RAM to at least 8 GB would be better
Change battery


That would be the things I did and now it can be use for daily but not extreme usage. Hopefully it is helpful as I just wanna share as it is the same device.
Additional video: https://youtu.be/83_GiAViMHo