Learn Hardcore Japanese: haha ha isogasii desu ga, keeki ga daisuki desu.

は is the topic marker. It sets 母 (“mother”) as the topic of the sentence (“As for mother…”). If you used が here, you’d be marking 母 as the grammatical subject in a neutral way (“Mother is the one who is busy”), but using は gives a broad “As for…” sense and often implies contrast or emphasis.

Why is です added after 忙しい?

忙しい is an い-adjective that can stand alone (“busy”), but when you want a polite statement you append the copula です. So 忙しいです means “(she) is busy” in polite speech.

What does ですが mean? Why attach が to です?

In ですが, が is the conjunction meaning “but.” When you attach it to です, it becomes “is…, but…” (Busy, but …). It links the first clause to the next one.

Why are there two が particles in this sentence?

They serve two different functions:
1) In ですが, が is the conjunction “but.”
2) In ケーキが大好きです, が is the subject/”object of liking” marker. With 好き/大好き, the thing you like is marked by が.

Why is ケーキ written in katakana instead of hiragana?

ケーキ is a loanword from English (“cake”), and most foreign words in Japanese are written in katakana.

Why is 大好き followed by です rather than だ?

大好き functions like a な-adjective (or noun) with the meaning “to love” or “to really like.” In polite speech you use です. In plain speech you would say 大好きだ.

Can you use を instead of が with 大好き? For example, ケーキを大好きです?

No. With 好き/大好き, the thing liked is marked by が, not を. So ケーキが大好きです is correct.

Is it possible to omit です for a more casual style?

Yes. In casual speech you can say:

母は忙しいが、ケーキが大好きだ。 (plain form with だ)
Or even more colloquially: 母は忙しいけどケーキ大好き！ (drop copulas entirely)

Why is 母 used here and not お母さん?

In writing or when referring to your own mother, you often use 母 (はは). お母さん (おかあさん) is more polite/respectful and commonly used when addressing someone else’s mother or talking directly to your mother.