The best thing to do here is to extend the life of your chair by contacting Whittemore-Sherrill and see if they will make you a set of armcaps in leather for the piece. I've done this before for a customer who had the same situation you are experiencing and that was by far the best and cheapest solution (assuming the rest of the chair is sound). If you have the model number of your chair and a swatch to mail to them they should be able to make you a set of armcaps in leather for $ 125 to $ 200. Make sure to tell them to use best available match, it does not have to be exact (otherwise they may decline the job).
Repairing a hole that size is not practical or easily do-able to any degree of satisfaction. Nor is recovering in leather, you'd be shocked at the cost (would be more then the price of an all-new chair)
Good luck!
Duane Collie
Straight answers from thirty-six years in the business.
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