A trick for next time (which may mean THIS time, actually.)
If you bring in one key to the shop for gluing, don't bring ONE key.
This may work, if you can un-glue the culprit.
Bring in the entire action (best way) or a group of keys surrounding the broken key. Take out the group and lay them together, including the broken one. Support them as necessary to get the group fitted to each other.
Restrain the group at each end, back ends and front ends, also.
Dry fit the culprit. Lay it into the group.
Examine. Recheck dry fit.
Does it fit correctly end-to-end?
Does it space evenly along the keys to each side?
Is it lying flat with its neighbors?
Is it against both the back stop and the front stop restraints you set up?
Plan a means to hold the culprit after gluing. (use paper, wax paper, spacer cardboard, etc to space it back into the group)
Apply the glue and your plan. Even stretched masking tape can provide enough pressure for the job.
Lay the glued culprit back in and triple check.
Your plan can even include lightly pressing the grouped keys against the culprit to assist in the final set-up. The point is to get it right, NOT tight.
Let it dry. Apply reinforcement to each side of the break, after the primary break is cured, if you choose to do so.
Note: It is even possible to do this on-site, after you un-glue the key in the shop ahead of time and let it dry out, etc. Using CA and the same techniques, you can fix it there.
(Pray for the Lord's help, of course, if you decide to glue a key with CA on a Steinway D. What could go wrong?)