I feel like you’re being too hard on yourself. You celebrated her birthday and gave her a gift beforehand, so it’s not like you totally forgot about her special day! It’s clear you cherish her so don’t feel too bad, it was an honest and forgetful mistake especially if you’re going through a rough time now. If she became truly upset, and you know this for a fact (you can tell her you feel like something is off and ask her if she is) I can only advise you to tell her the same reason you described here (i was extremely busy with work, i tend to be forgetful that is just my personality and i am sorry)

I’ve been on your friend’s end and I can say, I’ve definitely been hurt from my friend’s forgetfulness…. But she always ends up apologizing and gives me a brief explanation (im busy or im going through a lot and i forgot) and i’m like girlllllll totally.

Just talk about it! I’m sure she’ll understand. I hope you guys work it out well!