Killed somewhere around ten, but the mission completion doesn't budge, even after using the artifacts they drop..
Do I have to complete a round before the artifact's show up, or am I glitched?

you need to complete one step of the mission before the artifacts work.

What Vekno says is true. You need to start the mission manually, then the Anti-Cryptex artifacts will work. However, please note... During the first completion, you can use up to 9 artifacts. The second completion only 8, third--7, and so on. Therefore I recommend not doing the mission past 2 or 3 completions. The reward just isn't worth the energy spent beyond that. Plus when you hit rank 350 and Flamehawks show up on your NPC tab, you'll want the Data Analyzers around to lower the Flamehawk rate.