What you need to do is not to confront the director, it's to find another job.
You cannot work 14-16 hours per day. It's unsustainable. Getting sleep/rest/relaxation is known to be medically important; lack of rest can literally kill you. And I don't mean in the "I'll go crazy and commit suicide" type, I mean in the "contribute to severe illness which winds up in hospitalization and death" kind. You are literally killing yourself by doing this, so stop. If that means the work doesn't get done, then the work doesn't get done. Your company can fire you and find someone else who is literally willing to die for the company, because you shouldn't have to (they shouldn't have to either, but there's a sucker born every minute, as they say; don't let that sucker be you). The same goes, by the way, for working 7 days a week; even if you can get your workload "down" to 8 hours a day 7 days a week, that's still not good enough. It's simply unsustainable.
Here's another way to look at it: As a software engineering manager, you're probably salaried, meaning (in some places in the world), you don't make extra money for working overtime. Your salary is based on a 40hr work week, because that's (in most locales) the statutory definition of a full-time employee. Let's say you make $150k/yr as a software engineering manager. That works out to roughly $75/hr for a 40hr work week. If you're working 15 hrs per day, 6 days per week, you're making $33.33/hr, less than half. This works out to roughly $67k/yr, as a 40hr/wk job. To put this into perspective, you are making the same hourly rate as a junior fresh-out-of-college developer at an early-stage startup (actually you're probably making less than that), and you're literally killing yourself for the privilege.
Here's another, other way to look at it: You're a salaried employee (again, presuming this), and you don't get paid for overtime (again, presuming this). Your salary is based on a 40hr work week. Let's say you make $150k/yr. This means, every 40 hours per week that you work is worth at least $150k annually (it actually scales up; the more hours you work the more it takes a toll on your health, so the more hours you work the more valuable those hours are, but I'll brush that under the rug for the moment). If you work 15 hrs per day, 6 days per week, that's 90 hours per week. That's (roughly) $185k/yr in extra hours you are working and not being paid for. You are freely donating $185k/yr in man-hours of labour to your company and getting nothing for it. Now, what do you think the likelihood of this situation is to work in reverse? Do you think the probability of your company just going ahead and donating to you, free of charge or service and no strings attached, $185k/yr, is a high probability or a low probability? Would they treat you as well as you treat them? I'm guessing they wouldn't; perhaps you should ask HR about this (you probably shouldn't unless you want to get fired, but you may want to ask them this on the way out the door as part of whatever exit interview process they have, if such a process turns hostile for whatever reason and you feel like lighting a 5-alarm gas fire on this bridge and don't care about losing whatever reference letter or whatever they may give you). They wouldn't do it for you; don't do it for them.
OK, so we're stopping working 14-16 hours per day, 6-7 days per week. That's not happening anymore. Now, what's going to happen is that someone is going to get pissed off because they can't abuse you anymore (and make no mistake, what's been happening up until now is abuse). This is why you have to look for a new job, because this is almost certainly going to get you fired; abusers (in any situation, be it corporate or personal) tend to get angry when their abuse stops being tolerated.
As for the current situation, this is simply a symptom of the ongoing abuse. They know you have 2 other high priority events going on, and they simply add another one onto your list as though it's nothing. They have no respect for you, full stop, end of story. There's really nothing you can do about this; you explained the situation, and you simply can't be in 2 places at once. Of course, you can ask them, as other answers have suggested, to give you a priority list, but I'd bet dollars to donuts that the response will be "it's your task, you're a manager here with responsibility, can't you figure it out?" or something equally ridiculous. In which case, if you try to fight harder, you may find yourself in a disciplinary hearing (abusers, etc); it's not worth the time or the hassle. Since you can't be in 2 places at once, and your job description appears to include "be in 2 places at once", it's time to admit that you cannot perform the requirements of your job and to find another job whose requirements you can perform. Let another sucker try to take your job.
In the meantime, while you're looking for another job, just do the best you can. You'll probably get called in for various disciplinary hearings or whatnot over the fact that you can't do silly things like being in 2 places at once, or you need to take 5 hours out of your 15 hour workday to eat meals, or whatnot, but just ignore those. Start the ball rolling on a new job and you'll be happy sooner than later.
For what it's worth, you may want to look into whether or not your labour code supports cases of what's known as "Constructive Dismissal", which is basically where the employer makes the employee so uncomfortable in the job that they are forced to quit, thereby alleviating the employer from responsibilities such as severance pay, unemployment insurance, and so on, that they would have had to pay in case of a termination. In some countries (including the UK and Canada, and also some others), this is illegal and the company can be sued. If your country has these statutes, you may want to contact a lawyer to see if bringing such a lawsuit is in your interests.