As you can see, the battery in my MacBook was swelling and had pushed the trackpad out of the case. Apple support said that my device was too old to service in store, so I should find an authorized repair shop. But I live 2-3 hours from the nearest options, so I asked my office IT guru if he knew of anyone local. He sent me a link to iFixit, and told me that I could try it myself and save gas and money.
He was right, and it was as simple as the tutorial suggested. Other than my shaking hands!
The tutorial was pretty straightforward, honestly.