After several failed attempts on the harder difficulties, I found out what I was doing wrong!
If you manage to rescue some of the zerg pinned down in the caves (the bonus objective), you'll receive more zerglings and some... queens?
Conventional wisdom dictates that Queens are strictly for use in macro and base management. After all, they're slow, and they have the ever-useful "Spawn Larvae" ability.
Except in the HotS Campaign, you have "Swarm Queens" instead. These look and act much like the multiplayer Queens, but with a couple of notable differences. First, they lack the spawn larvae ability. Additionally, they are much faster when not on creep relative to their multiplayer brethren, spawn creep tumor does not cost energy, and have "Rapid Transfusion" instead (smaller, heal over time, smaller cost, auto-cast).
As Swarm Queens are not considered a light unit (so no bonus damage from fire!) and, importantly, have the ability to attack air, they are the perfect unit to supplement your forces.
With their "Rapid Transfusion" ability, a small HoT that groups of queens will auto-cast on each other, you should be able to hold off most of the Domion's early assaults with a handful of queens, Kerrigan, and decent micro.
Unorthodox as it sounds, you are expected (and explicitly encouraged!) to use Queens to hold out until reinforcements arrive.