First, 全部 is an adverb here. Quantities and numbers are often expressed adverbially in Japanese, and it usually comes after the noun (see this). Second, using は after 全部/全員/すべて/etc in a negative sentence has an important role; it distinguishes no and not all.
- 全員が来なかった。 No one came.
- 全員は来なかった。 Not everyone came (although some came).
- その本の全部を読まなかった。(†) I read nothing of the book.
- その本の全部は読まなかった。 I did not read all of the book. (I read only partly)
- すべて正しくない。 Nothing is correct.
- すべて正しくはない。 Not everything is correct (although they are mostly correct).
† その本はまったく読まなかった (using an adverb) is usually common and better
Note that in the examples above, 全員 is used as a subject, 全部 is used as a (pro)noun, すべて is used as an adverb. I think this type of は should be seen as a contrast marker, not a topic marker.
And this also means your attempt, ケーキは全部とても食べられない, is likely to mean "I can never eat any cake", which is totally different.
一度に here means "at one time". It is another very common adverb and it modifies 食べる. 一度に and 全部 are often used together ("all at once"). The following four sentences mean the same thing.
- 一度にそのケーキを全部はとても食べられない。
- そのケーキを一度に全部はとても食べられない。
- そのケーキを全部一度にはとても食べられない。
- 一度に全部は、そのケーキをとても食べられない。 (unusual, emphatic word order)