Another way to go the the Lippo Towers if you want to spent a few days in Hong Kong and look around is to do the following. When you get to Hong Kong only buy a one way ticket on the airport shuttle. It stops in three places, Tsing Yi, Kowloon, and Hong Kong. Get off in Kowloon or whatever the second stop is, take the Holiday Inn shuttle bus, it's free. Get off at the Holiday Inn. Walk to the Chung-King Mansion. If you cannot find it ask someone, when you are near it don't worry someone will get in your face and ask you if you want a room. Say yes, check it out, if you dont like it go back downstairs and wait for another to jump you. Don't be too picky, they are much the same. I didn't see any cockroaches when I was there. The place is not bad, and it's cheap to stay there compared to a hotel. Around $150HKD a night, I think. You can then take the Tsim Sha Tsui MTR, which is right across the street from the Chung-King Mansion, to the Admiralty stop, once there go to EXIT no. 4. Madmax has explained the rest very well.
Go get your visa the first thing in the morning, then go up to Victoria's peak and hangout until whatever time they told you to come back. It's really close to the Lippo Towers. Just be careful not to get lost in the bridge, you will see what I mean when you go there. It's the most complex walking bridge I have ever seen. So after you get your visa just do the reverse back to Tsim Sha Tsui. After you have had enough of HongKong, take the bus across from the Chung-King Mansion, as I remember it was a red bus, A1 I think... It's written in English and you can ask the person you get your room from at the mansion which one it is. The bus is only $17HKD and you get a little tour of Hong Kong, I think it was an hour to the airport.
Remember before you leave Taiwan you need three tickets. To HK and back to Taiwan, and then from Taiwan to another place. If you go the the Visa office with out it they will reject you immediately! And you should pay an arm and a leg in HK for another ticket. Madmax covers it pretty well. When you get back to Taiwan if you can get your visa, then just go to the airline office in Taipei and cancel your ticket and get a refund. Anyway good luck and have fun!