Double tonguing is one of those techniques that always seems to need attention. No one ever graduates from DT-101, as it has a way of appearing on each year's curriculum and in each week's concerts no matter how long we've been in or out of school. So we might as well get good and used to generous doses of that boring but so necessary section in the Arban book - pages 175-190.
Whoever heard of having to begin a note with a K, trying to articulate from the middle of the tongue, half way back to the throat? No wonder our double-tonguing is so easily handicapped. But it must be done. The trick is to mimic the clarity of the T so that no one can hear the difference, not even you. Painstaking slow practice does work, and that just might be the fastest route to mastery. Ironic that speed is accomplished slowly. Sadly, few have the patience. After all, there is absolutely no joy in trying to pronounce a T sound with a K at a snail's pace. And who has the time?
I remember a colleague at Eastman who was so exasperated with his spastic DTs, that he went on a K-only binge! Absolutely nothing but K attacks. His goal was a massive tongue-strengthening program. Heroic intentions, but he quickly developed a severe stuttering malady and had to spend two weeks in the infirmary.
There is almost no trumpet rep that ignores double-tonguing. First of all there is Ravel's G Major Concerto for Double and Triple-Tongued Trumpet with piano accompaniment (as I call it), the humility check. Then there are those particularly awkward passages in which the 16ths start with the K, or weaker note, like the solo in Capriccio Italien.
Equally hard is having to go from double to triple instantly. Example: Pines of Rome. As soon as you turn the first page, having executed high speed multiple tonguing with flying colors, Respighi's mighty tongue-twister challenges the trumpet player's self-control, often rendering him flummoxed and perplexed. Ottorino seemed to say, "You think you're good? Try this!" For me, he usually had the last laugh.
The list is endless: solos, orchestral works, chamber music, brass quintets, bugle calls, etudes, etc. Multiple tonguing is inescapable, and likely yours needs improvement. It is one of those "unmusical" items that tends to get neglected because it isn't fun to practice. Avoiding the unfun stuff soon makes the fun stuff not so fun. Learn to master the boring stuff, so that your playing isn't.