Some linguist argue that 了 DO NOT indicates 'completion of an action'.
Instead RVC ("Resultive verb complements") indicate 'completion of an action', but we are talking about 了...
了 renders either part or whole situation factual, that is real. Please, notice that factual situation does correlate (most of the time) with being in the past, but it could very well be in present or in future (see conditional factuality in ex.1)
- If you do X, then It's all done.
Part (If you do X) of the situation is real but the whole situation is "fake", that is it's a conditional statment. If condition X is factual, then Y will be done, but Y is not done yet (not factual yet).
- I doing this thing now. Whole situation is factual.
There is not much difference in your example though:
(a) 上个月我去台湾了. Last month I have already gone to Taiwan. So my visiting Taiwan is a fact. Without (上个月) this sentence won't be translated with the past tense.
(b) 她上个月去了北京. Last month she had gone to Beijing, .... (looks like sentence hangs in the air, it's not complete, because you had said that A became fact before B, but you didn't provide B)