Swimming coach wants me in ballerina shoes...

Yep, think of your hip as a hinge from which your straight (or nearly so) leg swings.  You have runner's kick which means you don't point your foot and you bring your knees up.  When I want to focus on a specific form I repeat a short phrase over and over while I"m doing it.  For this I say, "The hinge is the hips, the hinge is the hips" over and over again while I'm swimming.  As far as pointing the toes, most people with a martial arts or gymnastics or dance background have very little problem with this.  Other people just don't have much reason to practice pointing their toes.  It sounds ridiculously simple, but it is something you need to practice.  Lay on your back and do a slight leg lift, maybe 4-8 inches.  Then focus on lenthening your legs and point the toes.  Really bend your foot into it.  Repeat.  Add a little scissors motion to emulate kicking in the pool.  Don't bicycle your legs like you would do for an ab workout, just scissors kick up and down while keeping your legs looooooong.

Good luck.

Frankly, I'd say your best bet is to stop kicking.  Focus on improving your balance, your rotation, and your catch.  These three will do much more for your overall time than your kick.

Don't worry about 'not fixing it' or being embarrassed.  There are some triathletes that don't kick (and others that only kick minimally).  I'm one of them -- it uses a lot of energy and doesn't benefit me very much.

Like you , I have a terrible kick.  So during races and most training, I try not to kick at all.  (When I do side-balance drills, I almost always use fins so that I can focus on the balance.)  Yet, over the past year, I've managed some significant improvements (to me) in my swim time by working with a coach and focusing those other areas. 

you have to kick.  There is a difference (BIG DIFFERENCE) between no kick and minimal kick.  No kick and your legs act like anchors draging you down.  Minimal kick will maintain body position and help with your rotation.

the fastest way to learn how to kick?  Go in the deep end and tred water.  Staying upright, start kicking hard and lift your hands into the air.  If your kick is good you will stay above water.  If not, you sink like a rock.

Yeah...but that's Canada.

I kid.

I am and always have been about the worst kicker on every swim team I've been on, but still was a decently fast distance swimmer (4:34 for 500 yards, 4:03 for 400 LC meters). It's all about ankle flexibility.

And a word of caution about trying to increase your ankle flexibility: it is good for swimming, but not so good for running, as it decreases your lower leg stability and that can lead to, among other things, a propensity towards shin splints.