Do you adequately work all your body appendages when you swim? OK, let me be more clear ... When you swim, are your legs propelling you as much as your arms? When you get out of the pool after a good workout you may be arm weary, but are you ever leg weary too?
Yes, it all depends on the stroke, but it never ceases to amaze me how many people barely use their legs to propel themselves through the water when in fact some of the strongest swimming muscles of a given swimmer may be in the legs.
While many swimmers are quick to stick a pull buoy between their legs, it is the rare lap swimmer I see actually using a kickboard to give him/herself an aggressive workout for their lower half.
Instead kickboards get the short shrift. In the rare case I see someone who does use a kickboard, it is generally only for a couple laps or to work on their form. Some even - perish the thought - stick their kickboard between their legs and use it as a pull-buoy. In the twelve years I've been swimming daily--outside of watching swim teams practice--I have rarely seen anyone use a kickboard to go more than a couple hundred meters, and NEVER have I seen anyone use one to the point of exhaustion.
There are reasons. Kickboarding can be a bit difficult. Even if one is in shape in the upper body, kickboarding at first can be painful. Making muscles work hard that are not accustomed to an aggressive routine, always is. But once you get past that, it's just like any other exercise.
You can ease into it. Start with a set of short sprints 1/4 to 1/2 length followed by a couple easy lengths, then repeat. After a couple sessions start making those short sprint lengths longer. Set a goal to fatigue your legs -- at least briefly -- in each swim session. A little effort goes a long way to building power and endurance.
So here's the bottom line. Most lap swimmers' typical swim workout is not sufficient to build all available swim muscles and the legs are generally the ones that get ignored. I suggest adding a dedicated period of time during the aggressive part of your workout (not just the warm-up or cool-down) to kickboarding. After a couple months of this, I think you'll amaze yourself with additional speed and endurance, and when it comes to that final kick to the finish, it will be you who may have the advantage.