Learn Turkish: Birden aklıma geldi. - It suddenly came to my mind.

Hatırladım is generally used when you actively recall something, often after trying to remember it or being prompted by someone. Aklıma geldi (literally "it came to my mind") is used when a thought, idea, or memory pops into your head completely spontaneously, which is why it pairs so naturally with birden (suddenly).

Is birden literally just the number one (bir) with the "from" suffix (-den)?

Yes, exactly! It literally translates to "from one," but functionally it is an adverb meaning "suddenly" or "all at once" (similar to the English concept of "in one go"). Another common word you could use here is aniden.

Why is the word for "mind" spelled akl- here instead of akıl?

This is a classic case of vowel drop. When certain two-syllable nouns (like akıl, meaning "mind" or "intelligence") receive a suffix starting with a vowel (like the first-person possessive -ım), the narrow vowel in the second syllable disappears. So, akıl + -ım (my) + -a (to) becomes aklıma, not akılıma.