So, where is the case study with the numbers? People here will be able to offer suggestions once they see the numbers. It does not need to be perfect, you can update it as you dig out more data.
How old are the kids? Those sport expenses are over the top. You should be putting that money in their college account if you want to set them up for a better life. Your poor wife must be spending half her life in that van taxiing kids from one activity to the other. What does she think about this? Is she on board?
We don't have a strict budget, but we rough out a spending plan at the beginning of each year and i track the expenses (most of them). Fixed expenses, fun money, anticipated big expenses like a new roof or tires, needs and wants. Don't react to situations, plan for them. I regard savings as a fixed expense though, it is a line in the budget.
If you are the breadwinner, you have to sort this out asap, it is a dangerous situation. You make a lot of money, that should provide for a very nice, comfortable life. We have similar sort of household income, but we both work, and i know that my part time income would cover our basic expenses if my HB lost his job.That, and a hefty emergency fund takes away a lot of stress and pressure. No mortgage, just the last bit of a silly car payment. When he was forced to take 2 weeks extra unpaid holiday our first reaction was yay, the kids don't need holiday care and you can clean out the garage. You want to be in the same position and at that income you can totally do it if you focus. You will have to make some drastic changes in the short term, but it will be worth it on the long term.
Post that case study and start tracking your expenses. Do it every night with the wife. No judgement, no arguments, moving forward. An excel spreadshhet will do for the start.