I spent one night in KL, it was enough not to want any more. Malaysia is cheap outside the city, if you have time to kill on the way to Oz then you could stay in the rain forest resort for not much cash. The backpacker's hostels are on the lower side of the river, all the expensive stuff is up the cliff.
Taman Negara, Malaysia National Park
From Taman Negara there is regular transport to the station for trains to Singapore (I'm guessing that's your eventual destination).
You can't take your bike there but I'm guessing again that it's packed in a crate.