Grafted eggs won't work, the bees will not use them. Grafted larva between 12 to 24 hours old are all that should be attempted to graft. A hopelessly queenless hive will then build out the grafted larva to queen cells.
Using a 5/16" dowel, you can dip it into melted wax multiple times to get a good thick coating of wax on 5/16 to 3/8 inch of the tip. place in freezer then the wax can be removed from the dowel. Using melted beeswax as glue, glue the made cells to the bottom of a frame top bar. Graft into these home made cups and place the newly created cell bar with grafts into a hopelessly queenless hive. The bees will build out queen cells for you. 10 days later, remove the queen cells and place into hopelessly queenless 3, 4, or 5 frame nucs. 14 to 16 days later, check for laying queens.
I've been keeping a few hives for the past 20-25 years, and no longer treatment free as I got tired of the losses.