I have an idea: perhaps the kingdom did come at the transfiguration - but only partially fulling the prophecies (e.g. Daniel 2:24). However, it left again. Notice Matthew 17:
1 And after six days Jesus taketh Peter, James, and John his brother, and bringeth them up into an high mountain apart,
2 And was transfigured before them: and his face did shine as the sun, and his raiment was white as the light.
3 And, behold, there appeared unto them Moses and Elias talking with him.
4 Then answered Peter, and said unto Jesus, Lord, it is good for us to be here: if thou wilt, let us make here three tabernacles; one for thee, and one for Moses, and one for Elias.
5 While he yet spake, behold, a bright cloud overshadowed them: and behold a voice out of the cloud, which said, This is my beloved Son, in whom I am well pleased; hear ye him.
6 And when the disciples heard it, they fell on their face, and were sore afraid.
7 And Jesus came and touched them, and said, Arise, and be not afraid.
8 And when they had lifted up their eyes, they saw no man, save Jesus only.
9 And as they came down from the mountain, Jesus charged them, saying, Tell the vision to no man, until the Son of man be risen again from the dead.
10 And his disciples asked him, saying, Why then say the scribes that Elias must first come?
11 And Jesus answered and said unto them, Elias truly shall first come, and restore all things.
12 But I say unto you, That Elias is come already, and they knew him not, but have done unto him whatsoever they listed. Likewise shall also the Son of man suffer of them.
13 Then the disciples understood that he spake unto them of John the Baptist.
Why did the disciples ask their question in verse 10 if they did not think that the kingdom had already come, but without Elijah coming first? And Jesus tells them that just Elijah has already come, but not fully (for the people rejected John), so Jesus had come, but not fully in the sense that the OT predicted (he came to die that time, not to rule); and similarly the kingdom had come (at the transfiguration), but not fully. The people had rejected John, and would reject Christ, so the kingdom only appeared briefly. And that is why only a select "few" were alowed to see it before they tasted death. This may also explain why the disciples thought the kingdom would be restored to Israel after Jesus' resurection, as mentioned in Acts 1 - they'd seen it come and go previously, because Messiah had to suffer first. So why hadn't it "come to stay" after the ressurection?