Sorry to hear that you and your sister can't come to agreement. It's really a shame that the broker your sister hired didn't gauge your interest in the property at the start, since you could have settled the commission matter at that time.
In any case, I'm guessing that the property you're buying is in Chicago. In our city, the custom is that the seller pays the brokerage commission, and the buyer's broker (if there is one) usually gets paid by the seller's broker under a "co-operative commission agreement." Under normal circumstances, the seller's broker would indicate the amount of the buyer-broker's commission in a Multiple Listing Service information sheet or other document.
Most seller's brokers work under an "Exclusive Right-to-Sell" agreement, which gives them sole authority to market and sell the property. I'm guessing that your sister hired the broker under such an agreement, which guarantees the broker a commission, even if the seller finds the successful purchaser for the property.
You don't say it explicitly, however, it seems likely that your sister also hired a broker who is engaging in "Dual Agency," meaning that the broker is handling both the selling and buying sides of the transaction. Under this circumstance, the selling broker keeps the entire agreed commission and does not split it with another broker. The seller pays this commission from the proceeds at closing.
So, if indeed these are the circumstances, that a single broker handled both sides of the transaction, in my opinion, you and your sister need not pay each other anything. She, as executor of the estate, hired the broker to sell the property. The broker marketed the property and found a ready, willing, and able buyer, you said. You then exercised your right of first refusal and purchased the property. The broker still collects the agreed commission, since the broker fulfilled the terms of the agreement by finding a buyer, who you rejected, and by assisting in organizing your purchase.
Under the alternative circumstance, if the broker had sold the property to the outside party, the broker would have collected the agreed commission from the seller (the estate), and you and your sister would divide the proceeds remaining. In the current circumstance, the broker is selling the property to you, and the broker will collect the agreed commission from the estate. You and your sister will divide the remaining proceeds after the commission, which probably are the same amount, unless the brokerage agreement states otherwise.
If you think your circumstance is different than the one outlined above, contact an attorney to sort things out. I do hope this is not necessary, since the attorney's fees could in a short time far outweigh the amount in dispute.
Don Pasek, CIPS, TRC, ADPR
Omniterra Real Properties