CMCs are made to carry the least magnetizing current possible, so they are unsuitable for this type of circuit. You need much too large a part.
Gapped ferrite cores are ideal. Anything wound on a gapped shape (U or E cores with some gap -- add shims if necessary), or a rod, dogbone or bobbin style, will do nicely.
Or use an internally gapped material, like powdered iron toroids.
The inductivity (inductance per turn^2) will be relatively small, which means you need relatively many turns for a given operating frequency. That's fine, 50 turns of hair-fine wire (enough to power a single 20mA LED) is easy. Use a light touch.
If you want to make a bigger one, you can do it like this,
That's a #6 powdered iron (permeability = 35, reasonable losses). Current draw is about an ampere (from a fresh alkaline or charged NiMH), running the LED at about 1W.
If you're finding a lot of toroids with one color on three sides and another color on the fourth, try looking it up here,
http://www.micrometals.com/material/pcprop.htmlor here
http://www.micrometals.com/materialchart.html#26 and #52 are the most common; they're all but useless for this application, because they have very high losses. #3, 8 and 15 are the best.
More likely, you'll find an SMT ferrite core with a bobbin construction, which will be tedious to rewind, but most suitable for the purpose.
Tim