I asked this question awhile back, and the general consensus was that there isn't a good way to replace an eyelet that had shaken or broken loose. They're riveted in place on sides of the rim.
However, the outside "ring" doesn't really matter except to hold the eyelet in place, from falling out of the rim. When your wheel is already built, the only part that matters is the inside part of the eyelet, to distribute the spoke-tension load over that area of the rim. And the spoke tension will hold it in place.

It sounds like, in your case, there's no salvaging the eyelet and you just replaced it with a washer.
This is fine if the broken eyelet was for a non-drive-side spoke. (in nearly all traditional spoked rear wheels, the non-drive-side spokes have notably lower tension than the drive-side spokes). If it's for a drive-side spoke, there's a lot more tension to be distributed over the rim and I'd worry about the rim cracking. In this case you could rebuild the wheel by flipping the rim around so the broken eyelet-spot (now with a washer between spoke nipple and rim to help distribute the spoke tension) is now matched with a non-drive-side spoke.

I may be misreading you - in that case, could you post a picture?