I was worried this project was going to be even harder than it already looked but as it turns out Home Depot actually had the part we were looking for and it just took some assembly.
Chris simply unscrewed the old part, screwed in the new part and put it back together. Unfortunately we also have a leaky kitchen sink. The cold water tap drips a lot and unfortunately that part is not at Home Depot or any other independent plumbing parts business.
At least the main leak is fixed! And it worth the DIY, costing us only $20.00.