They last until you complete a long rest

Looking at the general rules for Temporary Hit Points (PHB, pg. 198), it says the following:

Unless a feature that grants you temporary hit points has a duration, they last until they're depleted or you finish a long rest.

Since the Storm Aura's Tundra option doesn't specify a duration, they will remain until the general rule says they disappear.

This is different to how Rage features usually work, but it could be thought of from a flavour perspective as the Rage was what summoned the "icy spirits", but the temp hit points come from the spirits that are now summoned, rather than from the Rage itself.

I'd like to begin by noting that multiclassing is a dangerous choice, especially for a new player. If you choose any single character class, the 5e rules will generally guide you to a reasonable character that is viable in combat. If you multiclass, it's possible that some of the levels you take will be "deadweight" levels which won't help you significantly in combat, and your function in combat will be that of a lower-level character.

In particular: you've told us that you'd like to play as a sorcerer. As a sorcerer, your function in combat will be to cast spells, and your fighter level will not make you any better at that. So, although your survivability as a multiclass character will be higher, your damage output will be that of a second-level character in a third-level party, and that's a significant drawback.

So the general advice we have for players of D&D 5e, especially for inexperienced players, is to not multiclass.

Let's now talk about details.

• As a third-level sorcerer, you would have six spell slots, two of which would be second-level. Good second-level sorcerer spells include invisibility, misty step, scorching ray, and shatter -- your four spells known could include two of these. Your AC might be 15 with mage armor or 12 without. When not spellcasting, you'd probably use the fire bolt cantrip, which deals 5.5 damage on average.
• As a third-level eldritch knight, you would have only two spell slots. But you would have a good weapon attack, you would have heavy armor, and you could use Action Surge, once per short rest, to attack twice. You might have AC 18 and average ten damage per attack (two-handed sword), or you might have AC 20 and average 7.5 damage per attack (longsword).
• As a sorcerer2/fighter1, you would have three spell slots and they would all be first-level. When not spellcasting, you'd use the fire bolt cantrip for 5.5 damage. But your AC would be 17 without using a spell slot, and you'd have the fighter Second Wind feature for 6.5 bonus hit points on average.

I don't think multiclassing serves you well here.

Now: third level is an unusually bad level to give up to multiclass to fighter, because the sorcerer gains a lot from their third level -- they double their spell slots and unlock a new level of spells. But fourth level is unusually bad too, because the sorcerer gains a lot from that one -- they get +2 to Charisma, which grants +1 to-hit and to save DC. Fifth level is especially terrible because the sorcerer gets fireball then, which can alter the course of whole combats.

What I'm saying is: giving up a level of sorcerer will hurt you a lot.

I'd like to idly note: if you did want to improve your AC, you might also consider multiclassing to cleric. Most of the cleric domains will grant you proficiency with heavy armor -- you wouldn't get that multiclassing to fighter. Cleric will also grant you spell slot progression, so a sorcerer2/cleric1 would have the full six spell slots including two second-level spell slots. It's almost worth it, except that cleric doesn't grant you progression in sorcerer spells known, so you'd still be a level behind in most of the ways that are important.