Try a good enzyme-based non-caustic drain cleaner like DrainCare. It won't harm your shower, drain pipes, etc., but it will eat out organic hair, gunk, soap scum, etc. that typically clogs tub and shower drains.
Run warm water down the drain, pour in the DrainCare, let it eat overnight, and then flush it down with hot water (I start that with a kettle of boiling water). If it doesn''t work the first time, do it again overnight until the drain clears, if it's a really bad clog.
Do NOT use caustic drain cleaners like Drano.
I think the reason the tip bent is because of hitting an elbow, like Mike, I don't care for the drano type cleaners, you can try putting your snake in slower, when you hit the blockage, just keep hitting and pulling back some, I'm not a plumber but don't figure you could go smaller then a 1/4 snake and the drain is at best 2" pipe before hitting the main trunk. Me, I'd use the plunger. good luck