How does *Steel Wind Strike* interact with *Heated Body* special trait?

Eddy the mage is facing 3 Azors, which are 15, 25 and 10 ft far away from him. Azors have the special trait Heated Body (emphasis mine):

Heated Body. A creature that touches the azer or hits it with a melee attack while within 5 feet of it takes 5 (1d10) fire damage.

Eddy casts Steel Wind Strike (emphasis mine):

You flourish the weapon used in the casting and then vanish to strike like the wind. Choose up to five creatures you can see within range. Make a melee spell attack against each target. On a hit, a target takes 6d10 force damage.

You can then teleport to an unoccupied space you can see within 5 feet of one of the targets you hit or missed.

hits all of them, and he does not teleport from his current position.

Since a melee spell attack is a melee attack, does Eddy take the fire damage given by the Heated Body trait? Is he considered to be within 5 feet during each attack?