It is interesting how the western mind, or maybe the human mind in general, cannot resist the temptation to formularize or pigeonhole God. The question, to which this paper is a response, is looking for the notion of a 'Right Time' in which God simply must answer our deepest and most heartfelt prayers. Yet a deep look at the passage will show us, I believe, that if Daniel had been asked to answer this question he would have said that he really didn't know that answer. He simply cast himself upon the mighty God in a genuine and unaffected way and it was that simplicity of Spirit that actually produced the hoped for (not required) response.
The first thing to notice is that Daniel enters this chapter after a long-term state of agitation. Back in Daniel 8, before the Persians had taken Babylon Daniel had already fainted under the weight of the visions he had seen. At the start of Daniel 9 we see him fasting. We are not told of a specific response to the vision in Daniel 9 but some two years later he was still sufficiently perturbed that he would fast for 3 whole weeks. If Daniel grasped the meaning of 'Messiah shall be cut off' then it is easy to see him being in anguish.
However, there is a difference between Daniel 9 & 10. In Daniel 9 we see him seeking the Lord's face and an explanation, in Daniel 10 we see him mourning. Generally you mourn for something that you accept is past. We don't see him actively seeking the Lord's face. In Daniel 9 he had been sent Gabriel to tell him what was to come to pass and he considered himself told. God had answered his prayer; he was not looking forward to any new or greater favor. He was simply expressing to his God his feelings and grief concerning what he knew would come to pass. Note specifically that we are not told Daniel was praying.
Verse 5 then lands us with an unexpected and largely unwarranted revelation: probably of the Lord Himself. Daniel is going to see more truth, but before he could accept that, he needed a greater vision of Christ. Whilst there is a perfectly reasonable physical explanation too (that Daniel was looking towards the ground) I find it very suggestive that Daniel has to lift his eyes up. This is very similar to Isaiah 6:1. If you look around you, and consider your lot in life, and your conditions then you'll have many reasons to mourn. What we have to do is stop looking in and start looking up.
Verse 8 is then, for me, a key verse. It introduces two things. First the notion of strength: Daniel comes back to this repeatedly in the chapter. Secondly: his comeliness had turned into corruption. Daniel had come to an end of his considerable personal resource and placed no value upon his own considerable merit. Before God can manifest his strength we must manifest and accept our own weakness.
Verse 12 gives us the element of timing. It is declared that Daniel's first movement towards understanding God's purpose had been immediately heard. It is also declared that it has been acted upon. Verse 13 suggests that the action had been immediate. Daniel had been mourning 3 weeks and the angel had been delayed 21 days. So, assuming that Daniel 'setting his heart' and his 'mourning' started at the same point the angel had been dispatched upon Daniel's first movement.
So may we assume that God is compelled to answer prayers immediately? No.
Why? Although he initiated a response immediately he allowed a 21-day delay to intervene. We cannot allow that God was forced to insert that delay, as He is omnipotent. Thus we must accept that God allowed Daniel to wait and therefore, at least implicitly, it must have been part of His will.
So, is there anything we can learn for our own prayer lives? I suggest yes, but it is probably not the answer the question is looking for. I think the first thing to learn is: take life seriously. If bad things are going to happen then we cannot resist them but equally we should not become inured to them. Secondly: we have to come to an end of ourselves, whilst we are still seeking to act God probably won't. Thirdly: God may delay a response, we may or may not ever find out why but if we have truly thrown ourselves upon Him then we will accept it.