Nope, I made a mistake there. It's good practice, when copying code from software with a different license, to call out what code is copied from where, but such a thing is not mandatory.
I'd say one of the things you have suggested. Copying the license file from spegel into a SPEGEL_LICENSE file in the repository would be sufficient. So would be actually crediting the project properly in the README with something like "portions of this code were taken from the Spegel project, under the MIT license" with a following copy of the MIT license.
