Try whacking the tops off one of the patches (if they are emerged out of the water enough) and spraying Round-Up on the cut portions of the weeds. Try another patch without whacking to see the effectiveness. Give it a week or so to work, compare and if the patches die, do the rest. If those are very slick leafed weeds, the round-up would probably just run off, hence the weed whacker. Round-Up should kill the root system. I'm assuming they are attached to the bottom as you said you are gonna dig/pull em up. I'll get some responses that Round-up (or generic) isn't registered for water application but Monsanto's Rodeo is, and is the same formulation as Round-up, glyphosate, just a little weaker.
If you wanted to take a picture of the weeds (close-up as well) and post here, I may be able to tell you what weed it is and give you the best herbicide for the job. If they are submersed weeds (underwater) another herbecide, like endothall or diquat would be necessary.