Hi Yoong,
>>The
children's mother
had been looking after them, but since she passed away, their
father
has / has had to look after them.
I don't like that past perfect. Without any context, I see you are talking from a present point of view (now, present situation, the father has to take care of the children). I'll modify that sentence in order to use "has" or "has had":
The children's mother used to look after them, but (ever) since she passed away, their father has had to look after them. - I don't know if it is better to include that "ever", it's just that "since" might always be mistaken for "because", especially when the clause with "since" comes before the one with the present perfect.
It was the children's mother who used to look after them, but since (= because) she passed away, now it's their father who has to look after them. - Here "since" means "because of the fact that", and so I used "has", simple present. This was my first interpretation of your original sentence, because I always tend to consider "since" as "because" when it comes out of the blue.
That's my opinion, not sure it's correct though. Let's wait for someone else.