She mentioned that one of the previous computers in the lab cost about $10,000, but that it had been a pre-built machine and I mentioned that for $10K I could probably do a LOT more.

Risk vs. reward. I value this experiment as 5/5 risk level considering you don't know what you are doing. I value the reward as 1/5 because if it works right nobody will care in 6 months. I'm not trying to be mean or offensive. I've trying to convey the seriousness of the situation. Use u/danplaysvgames build below but consider the following before you submit your parts list.

The first step you need to figure out is what parts this other 10k-machine has and when it was built. You will probably find 6k-8k in parts. Now you need to figure out what software is running and how this 10k-machine stores data. Is it backing up the data? Is it raiding the data? If this special software only uses a single core out of the 16 CPU cores you are wasting money. If this 10k-machine is raiding data you will need to figure out how to raid data and how to replace harddrives. The harddrives you have picked out can store a lot of data but are susceptible to high(er) failure rates. Worth considering if you have precious data.

Anyways, I could ramble on for a while but here are your steps.

1) Make sure your build is physically equivalent or better to the first machines.

2) Make sure you can setup the software to run correctly 100% of the time.

3) Make sure you can setup any security that was on the first machine.

4) Multiply whatever amount of time you think you need to setup this machine by at least 3x. It's better to come in under budget always. I'm 99% sure there is nothing fancy going on but there will be some learning in the future days OP.