I'm just nearing 50, and am seeing physiological changes in me that are moving faster than my mental capabilities allow. That is to say, my mind is saying "go!, Go! GO! Faster!", and my body is saying, "Hold on, there cowboy..."
So age will creep up on you to the point that your training methods will make a difference. My Taekwondo instructor tells me I have a typical American hips. A sedentary job, mild workout on the weekends, coach from the sidelines, drive to the supermarket even if only 2 blocks away...
That's not exactly me, but that's fairly common.
As stated before, dynamic stretching is definitely the way to go. You can, instead (or in conjunction) do PNF for faster results, but PNF can be more difficult, especially as it often requires needing a partner.
With dynamic stretching, you can work out as many times a day as you want, without giving any rest days in between, as long as you are not otherwise injured. In fact, with the best dynamic stretches, you begin a routine within 15 minutes of waking up in the morning. The nice thing about dynamic stretches is that they really aren't stretches per se, because you aren't going beyond your normal range of motion. However, you are keeping the muscles active, and that is most important in anything you do.
Once you find a routine, keep at it. Meantime, I wonder what you mean by "not good enough for TKD". I suspect you're worried about competing with the 18 year olds in class, and that you're in a sport school. I've got a problem with that, because that may be how Taekwondo is taught these days, but that's not what Taekwondo was originally taught. I've got students in my class who can knock a can off the top of my head. But I can crack his ribs before he lands the kick. He may have snazzy kicks, but I'll be the one to survive. And that's what Taekwondo was originally all about: survival, not some grunge competition to see who can kick the highest or with the most turns in the air.
And that's where I'd suggest you start your focus: forget the jump spin kicks. Flexibility is only the beginning of your issues. You risk spraining a knee, or an ankle. Or you can tear an ACL. You need to decide why you are in Taekwondo at all, and then train for that.
If you're there to do jump spinning high kicks, I'd say at 44 that time has long past. You might do okay, but at what risk, and at what cost, and for what benefit?
If you're there to keep in shape, I'd say you're not in the right place. Whether you're there for sport or for self-defense, Taekwondo is dangerous for keeping in shape. Keeping in shape does not mean risking ankles or knees, or bloodied knuckles. It means maintaining flexibility, mobility, your heartrate, your blood pressure, your cholesterol, your diet, and your mental faculties. Taekwondo is about self-defense (or trophies, depending on the school you land in).
If you're there to learn self-defense - and the school is competent - then you need not worry about how high you can kick. You don't want to kick high anyway. If you look at old footage of the original pushers and movers of Taekwondo, you'll note their kicks rarely hit stomach high. Here, your focus is in applying techniques according to what you can do. If you struggle to kick high, what good do you think that's going to do you in self-defense?
The trick here, though, is to find a competent school. Admittedly, this is hard to do, and is why so many people ask silly questions like "I'm 18, is it too old to start taekwondo", or "How high should a 40 year old be able to kick". If you're in it as a sport, then these are sound questions. If you're in it for self-defense, these questions are moot. If you're in it for keeping in shape, then go buy a Billy Blanks DVD and stay in the basement.
Therefore, to answer your question, you need only perform dynamic stretching, and daily practice each of your kicks and forms. You might not have a partner or a kicking bag around, and that's okay. Just do what you can safely at home, spend about 15 minutes of kicking (after your 10-15 mins of dynamic stretches), for a half-hour of practice each day. Don't spend too much time kick practice, you really do need a partner to help correct technique; otherwise, bad habits settle in that make it difficult to correct later.
Incidentally, I'm a huge fan of Tom Kurz and his published materials. He's an accomplished coach and author. He explains muscle physiology and optimal exercise regimens. Read his work to understand the best stretching methods you can employ. (No, you do not need to suspend yourself across two chairs as in his book and DVD photos; however, if that's what you aspire to, follow his routines TO THE LETTER.)