Do you offer monthly rates?

It's really up to you - they probably aren't going to find anyone much lower than $16/night. That's already a very low rate. Yes, the dog will take a slot that could go to a higher paying guest, but what are the chances that you would have another boarder every night this dog would be present? If you are normally fully booked nearly every day of the week, then I would hesitate to discount. If, however, you aren't usually fully booked, then the likelihood that you will lose money by giving a discount is slim. You can do the math for each potential discount to see how many nights you would need to have another full priced dog before you'd show a loss in profits, or you can just go with your gut. Sometimes giving a break to a sure thing (and making a grateful customer) is a better practice than holding out and hoping for other requests.

Personally, I find longer stays easier than shorter ones, excluding daycare. The dog usually settles in after the first few days and really falls into the routine. They become more like a family member than a guest. Multiple shorter stays, by comparison, you rarely have a dog really get settled in. Even if they're very adaptable, there's always an adjustment period, and I'd rather have one dog for a couple weeks straight than 3-4 dogs over the same time period, because that adjustment period is the hardest time. So for me, given their circumstances, I'd be inclined to work with them at least a little, even if that's just rounding down the total to the nearest $50 or so.

Oh, just tell them that you were planning to raise your rate but will keep it at $16, which is lower than what other local sitters and kennels would charge. You are 36% cheaper than the sitters in your area that charge $25/night. That's a huge discount.

I would definitely increase your price for the sole reason that you are loosing money and potential future clients. I actually do the opposite for that reason - for stays over 3 nights, I increase my rate $5.