Unscheduled Home Improvement: Problem

I have a loooong list, as I previously mentioned, of home improvement daydreams that I'd like to accomplish. Occasionally, something happens to make a "want" a "must." Such a thing happened about two days ago. I came into the kitchen to do one thing or another, turned on the kitchen faucet, and nothing happened. No water came out. Instantly, I was seized by an irrational panic that our water had been shut off, and a mental reviewing of myself mailing our bill most recently. I knew I had paid the bill, so it couldn't be that. Breathe, Self...breathe. Just to double check, I turned on one of the faucets in the downstairs half bath, and water came out, just as it should. So, I asked The Husband if he knew that the sink wasn't working.
"Oh, yeah," he said. "I turned the water off because something on it broke, and water was shooting everywhere."
Awesome.
We tried a few things but nothing seemed to work. Upon closer inspection of the sink, we could see the hole where the plastic "washer" had broken...and assumed that's what was causing our domestic monsoon.

Have I mentioned that this week is the week of Backyard Bible Clubs for our church, or that my husband is the Associate Pastor, and over the Youth Group? Well, I'm mentioning it now. That meant, essentially, that I would be on my own for this project. The Husband was going to be very busy this week, and wasn't going to be readily available for renovation projects. I'm relatively handy, but this was beyond anything I had previously attempted. Could I fix it?