First a small disclaimer: I'm not a judge. This answer is combined from information I found from Platinum Angel's rulings, comments of MTG judges from the Internet and common sense. For example a head judge of a tournament can turn over this ruling. This IS NOT an official ruling.
You will lose the game. Platinum Angel only prevents you losing from game effects.
Rulings on Platinum Angel:
1.10.2009
No game effect can cause you to lose the game or cause any opponent to win the game while you control Platinum Angel. It doesn’t matter whether you have 0 or less life, you’re forced to draw a card while your library is empty, you have ten or more poison counters, you’re dealt combat damage by Phage the Untouchable, your opponent has Mortal Combat with twenty or more creature cards in their graveyard, or so on. You keep playing.
1.10.2009
Other circumstances can still cause you to lose the game, however. You will lose a game if you concede, if you’re penalized with a Game Loss or a Match Loss during a sanctioned tournament due to a DCI rules infraction, or if your Magic Online(R) game clock runs out of time.
Normally Platinum Angel would prevent you from losing to state-based actions, but it appears that the "sudden death" SBA is special.
Best explanation I could find to this is in this post: https://apps.magicjudges.org/forum/topic/32928/
In the single elimination portion of a tournament, what happens if a player would lose to the “sudden death” rule while controlling Exquisite Archangel?
My understanding is that they would still lose the game, because the sudden death rule is akin to a Game Loss in that it bypasses all the rules from the CR (in the same way that a Platinum Angel cannot protect you from sudden death), but I'd like confirmation.
Your understanding is correct. As with its predecessors, Exquisite Archangel cannot prevent a player from losing to sudden death rules in a timed single-elimination match.
