Some pretty good responses so far, so I'll just add a bit of info on technique.
When a wave builds upon a submerged rock, it builds quickly and then dissipates on the other side of the rock into a gentle swell (unless conditions are rough). You have to lean the kayak into the wave, brace with the paddle(low braces work fine in this situation and are much preferred by this kayaker since the risk of injury is low). If the kayak broaches, as you suggested yours did, the full weight of the water will be on the spray skirt as you lean into the wave.
A skirt of the size required to cover such a large cockpit would, most likely, collapse under the weight of the water. Even though the time the skirt would be under stress is short, the forces would probably exceed the friction that holds the skirt to the boat. Any skirt that stayed on would likely be very hard to release if you had to exit the boat for other reasons.
For this reason, if there isn't an issue with storing gear in the bow, a large float bag up there would help limit the amount of water taken. This may even be sufficient to avoid making a wet exit the boat if the cockpit is swamped (though stability may be compromised).
I once took a 2 person large cockpit boat out during a demo day with my son in front (Coyote Point, SF Bay, 2-3 foot chop). I was really skeptical about the boat, but since we had lots of support and wetsuits, I decided to give it a try. The boat had little rocker and the first wave raised the bow insufficiently to crest the wave. As we passed through, the boat filled completely and we were essentially swimming. It was pretty funny, actually, since without a spray skirt, this event could not be avoided. A canoe would have fared better since the tend to rise up more than this barge did.
In this instance, the cockpit design was incompatible with the hull design and in waves (either less cockpit area and a skirt or more rocker up front), its performance envelope was easily exceeded. The boat, even in modest wave, was simply unsafe and could only be used in lakes or calm water - never on the ocean.
I'm not saying your boat fits in this category,
since I don't really know the performance characteristics of your boat (though I have some guesses based upon experience), but I will say that you should do some testing to find just what you can, or cannot do, with the boat before stocking it full of gear and taking it out in rough water.