Clogged Drain

The drain in my kitchen sink is slow. Water just lays there for minutes before vanishing. I can hear gurgling. I tried using a plastic snake, but it snapped at the handle and now that is stuck in the drain, further complicating the problem. I did some cleaning and rinsed out rags in the sink and now I realize that was a mistake. Dirt can be thick and accumulates. Plus I brush my teeth in the kitchen because I have a new bathroom vanity and I don't want toothpaste stains in there like my previous sink.
I will ask the super for the number of our condo plumber. Maybe the condo will pay for it. I wish I could just lift the drain cover off and grab whatever is blocking water. I also tried drain opener to no avail.
I'll tell you this--I will not stop brushing my teeth.