Programs: UA Premier Gold, DL FO, DL MM, Marriott Gold, Hyatt platinum
I twice went through TPE 18 months ago -- in first on MH. MH had no showers and I'm not sure they even had a lounge. I spent my time in pursuit of showers.
On the morning arrival from the US, I talked my way into a very nice CI (China Airlines) lounge with showers. I had the MH first ticket, elite status on DL and a DL Crown Room (now Sky club) membership. Not entirely sure which one made the difference. Also the lounge was nearly empty and showers were wide open.
On the return with an evening arrival, I was not granted lounge access. It was quite crowded. Eventually, however, I was directed to a public restroom with showers. Had to pay a small fee for soap & towel, and the place wasn't exactly sparkling, but I did get a shower before resuming the long trip home.
I'm guessing the OP wishes to shower before exiting the airport?
The EVA lounges are in departure area of T2, MH arrives in T1 (correct me if I'm wrong, somebody...). I don't think they will allow arriving PAX to use them, even if arriving with EVA (unless transiting).
More have lounges in both terminals... here is the link to the T1 lounge.
Like mentioned, CI has good shower facilities in their lounge. Not sure about the Eva lounge as I've ceased flying with them for ages but Eva do have an airport hotel within the terminal serving those passengers on transit, as far as I know, passengers on EVA and CI can pay a small fee to access them, not sure if they take MA PAX though.
there are public showers in T1 and T2 transit areas but you need to hunt them out EVA has a transit hotel in T2 but often full. If you are on MH and I guess your coming from KUL its hardly a long haul flight so use the lounge in KL or the departure point you are starting your journey from with MH