If it was so easy why didn't you fix it yourself. It doesn't matter what the final solution was, she had the knowledge to know the box needed to be checked. You obviously did not. You paid for that knowledge. Just how long or difficult would the problem have to be for you to think it's worthy of charging for and how do you judge how difficult the problem is? If the service person knows what they are doing, most problems are fairly easy. That's what you hire them for. It was obviously a very difficult problem for you.
I absolutely do not believe that the other companies would make a house call and if it takes less than 15 minutes they wouldn't charge you? NOT! That would be recipe for going out of business in a hurry.
A service organization HAS to have minimum charge. They would be fools if they didn't.