An Eldritch Knight uses wizard spells, therefore, taking a wizard Magic Initiate should (I believe) extend the ability to cast the level spell granted by this feat.
However, in the character sheet there is no way to cast it with level 2 spell slots (other than manually)
Edit: In fact he goes on to specify that an EK can use magic initiate spells from other class types the same way.
According to Sage Advice:
https://www.sageadvice.eu/2015/09/24/can-wizard-use-magic-initiate-feat/
http://media.wizards.com/2015/downloads/dnd/SA_Compendium_1.02.pdf
An Eldritch Knight uses wizard spells, therefore, taking a wizard Magic Initiate should (I believe) extend the ability to cast the level spell granted by this feat.
However, in the character sheet there is no way to cast it with level 2 spell slots (other than manually)
Edit: In fact he goes on to specify that an EK can use magic initiate spells from other class types the same way.
Playtesting Fugare Draconis, an epic tale of adventure, loss, and redemption
This is passed to the team. Keep you posted.