The following sentence is from this article on NHK Web特集
数えてみたら、息子と一緒に過ごせる時間がとても限られていることに気がつきました。
My attempt at translation:
When one tries to count it, they took care that the time shared with the son is very limited.
I'm pretty sure that 数えてみたら here basically means what I wrote above, meaning something along the lines of
If one tries to measure the amount of time...
But since it sounds very cumbersome in my translation, I wanted to ask for confirmation.
数えてみたら、息子と一緒に過ごせる時間がとても限られていることに気がつきました。
Your translation of 数えてみたら meaning "When one tries to count it" is spot on, although in this context it more likely means "When I tried to count it" — you misinterpreted the (implied) subject of the sentence, because I think you mixed up 気がつく "to notice / realize / become aware" with 気をつける "to take care".
数えてみたら、息子と一緒に過ごせる時間がとても限られていることに気がつきました
Trying to count it, I became aware of the fact that the time I was able to spend with my son was very limited
