It sounds like you have 2 issues going on. Correct me if I'm wrong, that's OK.
First, it sounds like you pulled the handle on the grey tank valve to dump the tank and when you did, the handle broke. Now assuming, just the handle broke (the plastic part), it an easy fix. You can simply unscrew the remaining shaft and replace it with a new one, sold at any RV parts store, any RV dealership, or places like Camping World.
If the handle and shaft both came off in your hand, then all you need to do is screw the shaft back on. These unscrew over time. If the actual valve is hidden under the under covering of your camper, you may need to cut a hole in the material to access the valve to find the opposite threaded stub to reattach.
However, if the valve itself actually broke, there is only one fix, and that is to replace it, which might require you hiring someone to do that.
If the system is a pull cable and the cable broke or snapped loose at the valve, then you'll most likely need to hire someone for help.
Second, it sounds like your sink is clogged. As stated above, bleach is not going to do anything. You need to know if your tank is full, dumping could take care of your problem.
If it truly is clogged, my guess is, the pipes are filled with grease and sludge. Caustic chemicals may, or may not work. Draino is caustic. Your RV pipes will not be hurt by using products like Draino, but you run a bigger risk of the Draino burning your skin or touching other materials in your camper that could cause damage.
What works better and safer is to use an emzine drain opener, found at any Lowe's or Home Depot type store. Once the emzines start eating the grease and sludge, they will clean out the pipes completely, without damaging anything, including your skin or anything else. FYI, Rid-X for septic tanks will not work in drains. It takes much too long.
About your drains, you should never wash greasy items in your RV sink, as the grease will easily build up and cause clogs. We always wipe everything with grease on it with paper towels first, then throw the paper towels away. .... always! without fail.
And because paper towels are so danged expensive, we switched to shop towels years ago, mainly because i do a lot of wood crafting and I use shop towels for varnish and stuff like that. Shop towels work great in the kitchen and are a LOT cheaper than Bounty, etc.
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or a bigger box
Wall Mart sells them and are a bit cheaper than Lowes. And yes, we use them for cooking too, like draining bacon grease, and wiping things down.
Good luck, and hope you can figure out your problem.