In theory, you can sell anything, it's all just a question of price. So yes, your business is sell-able, especially (!) since it turns a profit (!!!).
Some follow up questions:
You have four clients and profits are between $150-$180k. You charge each client $50k/year? Is that the cost for one autistic child? Who is paying that; the family or the gov't (military)?
Since the families rotate every three years and you have only four clients, on average you will lose 1.33 clients every year with little you can do about it. How is your out-reach to find new families/clients? Beyond word of mouth, of course. Has it ever been a problem?
I think that you've already hit on the main targets. Now you just need to go about doing it. If you have an employee who would like to take it over, that might be the best/easiest option. This person would already know the business, the clients, the seller (you!), the business model, etc. You would need to approach this person in advance and put it on their radar screen, with a window of 12-18 months (or so), especially if they don't have a business background. They'll need time to get used to the idea of owning their own business and making all the decisions, paying the bills, paying the other employees, finding new clients, etc, in addition to doing all the work they're doing now.
I'd make them a sweetheart deal, knowing that if they say 'no' and you close down, they could in theory pick right up where you left off. I'd saying something like 'X% of the previous 12 months income' (either pre-tax or post-tax, as you like), payable in monthly installments over the next Y months. I'm assuming that this person doesn't have a stack of benjamins laying around unused. Again, this isn't the best way to maximize the sale price, but you will be helping out someone else financially (presumably) and giving them a huge leg up in life (again, depending) as well as staying true to yourself and the kids.
The way to maximize the sale price is to find an outside buyer. This will take time and energy and may in the end be fruitless as in the end all that you have is a client list, rental space, employees and some equipment, no proprietary information, no product to sell going forward, etc. You may be able to get all the money up-front (hopefully), but these kinds of buyers don't grow on trees.