The slime remover won't remove the nutrients. The cyano may die back, but all the nutrients that fed it will be released into the water.
Siphinoing will remove the nutirents because they will be bound up in the cyano your sucking out.
Since you said your ref/cheato is starting to keep up with the nutrient levels, I would say your much better siphioning. That way the cyano gets removed, no nutrients are dumped back into the water, and the cheato will have a good chance of keeping up so the cyano doesn't come back, or at least not much.
Like fanagle said, any tubing will work. IME something around 1/2" ID is good. If you find it sucking up rubble and getting clogged, use your extra hand to slow the flow out of the siphion tube so it isnt' sucking hard enough to suck up any more rubble.