from Jan 21-21/ will or would

Instead of would like to ask, I would tend to use request. "Would like to ask" implies you want to ask, but not necessarily that you will or are. From the tone of your message, I get the feeling you want to suggest this person to fill your spot during your vacation and I feel request would fit nicely there.
Also, I can't comment on "from January 21 - 24" as it sounds decent, although I avoid using symbols in formal notes. If you want to avoid the situation, you may try "vacation leave beginning January 21" as the phrase "4 days'" implies the end date of the vacation.

At any rate, I hope you enjoy your vacation. =D

- PeckyP -

Sorry for the double post, edit disappeared...

Anyway, I just reread the second line and I would phrase it differently. Though, I have my own style of writing that I developed while I was younger.

Instead of "I would bring her up to speed on all jobs I am currently working on", I would say "I will bring her up to speed on all of my current jobs."

I change "would" to "will" because of the tense issue it causes. "Will" gives a stronger implication of the future act you will commit if she is chosen.
Also, I believe "on" is a preposition in this case and it is improper to end sentences with a preposition. So reworking it, I compressed it a bit while keeping the important parts of the phrase. (Or at least I hope I did. =P)

- PeckyP -