Sheepishly I've got to admit I responded before reading the portion of the text in Isaiah you mentioned in the OP, Paul.
In Isaiah 36-37, Hezekiah was praying directly to the LORD.
I don't see him asking Isaiah to pray (am I missing something?).
Isaiah 37:14–17 (NIV)
Hezekiah received the letter from the messengers and read it. Then he went up to the temple of the LORD and spread it out before the LORD. And Hezekiah prayed to the LORD: “LORD Almighty, the God of Israel, enthroned between the cherubim, you alone are God over all the kingdoms of the earth. You have made heaven and earth. Give ear, LORD, and hear; open your eyes, LORD, and see; listen to all the words Sennacherib has sent to ridicule the living God.
Observations:
1) Hezehiah prayed to the LORD
2) he prayed in the temple of the LORD
3) [
conjecture] if the king arrived at the temple the High Priest would probably have been notified.
4) [
conjecture] Isaiah was aware of the Hezekiah's prayers
5) The LORD responded to Hezekiah's prayers through his prophet, Isaiah.
6) Historians don't record every fact, only what is important to tell their story. Scripture likewise tells a story
but with a theological purpose and it doesn't tell us about everything. What was unimportant to the story was not necessarily recorded. We don't know who else was praying or who else was asked to pray - but I'd presume all the people knew they were in dire straits.
Rob