The reductive amination might be difficult just due to the close BP of acetaldehyde (21 C) and the product. You might be better off to do a simple alkylation of the ammonia with the ethyl iodide (bp ~70 C) as that would ensure allow you to distill the ethylamine from the reaction. You might get some di/tri alkylation, but that would be reduced by using excess ammonia, which is likely your more expensive reagent.
If you do the reductive alklyation, you would also need excess ammonia to prevent further overalkylation. Depending on the next step, you could react the ammonia with acetyl chloride making the N15-acetamide (note Aldrich sells this for $300/gram), which would be easier to isolate, then reduce with Borane or LAH to the amine, which could be then distilled easily from the rest of the muck, and then isolated as the free amine or reacted with HCl to store it.