Don't you think there should be a comma before "All while holding a baby"?
No, it's not necessary. The whole text is formed by phrases, not sentences, so you don't need to join those last two phrases with a comma.
Do "all while" and "all the while" have the same meaning?
Yes, you could also say "all the while" there. The diference is "all the while" is a noun phrase that works as an adverb (similar to "all the time"), while "all while" is parsed as two words: "all" refers to all the actions mentioned before, and "while" is a conjunction (all these things she did while she was holding a baby).