The language that you're referring to, where it states that if they do not provide responses to legitimately served discovery requests in a timely manner, that they would be responsible for attorney fees, this does not refer to your attorneys fees that you incurred in defending the suit. It refers to attorneys fees that would (actually could) arise out of a hearing on a motion to compel, in the event they never answered. If that occurred, the law allows you to ask the court to award you reasonable attorney's fees as well as sanctions, but only those having to do with getting the court to make them answer. Importantly, despite the rule that states this is a potential repercussion for continually failing to answer, they rarely get awarded. This usually only occurs when the court has already warned them, after you (i.e., your attorney) has willingly given them extensions, the court has given them further extensions, and they still failed to produce/respond. Typically an attorney will allow the other side substantial extensions of time, and this is something you may not even know about.
When you say they completely ignored the deadline, I'm assuming you mean the deadline on the discovery notice that gets served with the papers. Interrogatories and Requests for production of documents rarely get done anywhere near the deadline in the rules, which is a mere 3 weeks. Many times, it takes much more than this to track down everything that was requested. This is why extensions happen all the time and unless you're asking, this isn't something your lawyer will even discuss with you.
When you say they "didn't offer a remotely reasonable settlement until after the deadline and didn't finally dismiss the case until months after the deadline," I'm assuming you mean they didn't make a reasonable demand (it sounds like you were the defendant). This is actually very common, and in fact, it is very early on to make (or reduce the original) demand low enough that the Defendant will accept it prior to the discovery deadline and all the depositions have passed. (When I say deadline, I don't mean the one you're talking about, I mean the actual discovery deadline, which is set forth in the scheduling order; this can easily be 9 months from the time an Answer to the Complaint is filed.)
If you're referring to the token deadline put in the first set of interrogatories served, this not a "real" deadline anyone of the attorneys expects to be adhered to. Further, this a very quick settlement and you should be happy your attorney disposed of your case so quickly. As you pointed out, you are paying a lot of money every day the case lives on. In fact, the money you saved by settling early is substantial. If your attorney had gotten the documents and responses and had to wade through all of them, organize them, send follow up requests, take depositions, etc., you would be out easily another $10,000. Your lawyer did you a favor, because a less honest attorney would tell you to wait, to see if there is a defense, just so they could pad their bill.
Many times when it is clear that the case is going to settle, the lawyers will serve pro forma discovery, and will say to each other not to bother answering while they attempt to settle. They are timely served if you cannot settle, but it's clear that settlement is the ultimate goal. This is very typical when the defendant almost certainly has exposure, but when the plaintiff's case also has some holes. Because of issues on both sides, they agree a modest settlement to dispose of the matter, quickly, is the best course.
When you say the settlement explicitly involved each party paying their own attorney's fees, all settlements contain this clause. I have never seen a settlement agreement where a party agreed to pay the other's attorneys fees. It's just not done. In the rare case it is, it's part of the structure of the settlement and it flows to the Plaintiff, not the Defendant. This may occur in a civil rights case where there are no real damages, but the statute allows for attorneys fees to be awarded if even one-dollar is awarded. So, in a case like this, sometimes the plaintiff will accept their attorney's fees being satisfied as the settlement, (usually along with some sort of consent decree), so as to curtail the abhorrent behavior.
If you want to discuss these issues with your attorney, they are not billable: they are administrative pertaining to your bill; hence, you can do so without fear of being charged. If you're nervous, tell him ahead of time you'd like to discuss your bill. He won't try to bill you for this, as he can't, and furthermore your case is settled so your file is closed.
To answer your question explicitly: No - your fees are not recoverable. This is not only because you've already settled, but you were never entitled to them anyway.
To answer your question about the point of sending discovery at all if you are not going to expect answers and the goal is to settle, (1) is to preserve the right during the discovery period, in the event settlement negotiations break down; (2) to give the other side a picture of how sharp your attorney is, and that he/she will be asking the right questions and they will be invasive; and (3) this is just how it is done. What you've described is how almost all low level cases proceed.
Lastly, just to address what you said about it taking a few months from reaching a verbal or "handshake" agreement and having the settlement actually be recorded by the judge and a dismissal issuing, this is just something that takes a little while. Depending on the type of case it is, the court may need to approve the settlement. Even when it's not necessary for the court to approve the settlement, it takes a while to go back and forth on the language, draft the stipulations of dismissal and so on. A few months is right on target.
It sounds to me like you had a pretty effective and honest lawyer who could've dragged this out for much longer.
Advice for the future, in case you ever find yourself needing the services of an attorney again: If you have these types of questions, you should ask them as they come up. Again, it's not something that you can be billed for, and your lawyer should be happy to answer. Some lawyers are better than others in remembering to explain what the technical stuff means, and what the game plan is. However, the client has a responsibility too, which is to ask if you don't understand.