For the last "Who knows where" I found the mortar cannon to be EXTREMELY  effective. However in certain phases it falls flat, (The spouts that produce the green eyeballs and the elite melee guys) 
 
Does anyone know a secondary weapon that could dispatch these two specific characters easily? I dont want to get that far again to only fail again.

If you are doing a 10 idol run, I would suggest referring to my thread here. Ultimately, the mortar is one of, if not the, most powerful weapon in the game. I didn't have trouble killing the Stinkeye spawners with it, it just requires patience and good dodging skills. For the elite melee, figure out a good secret ability that will help you out. The last reflection is a doozy!

I love the spear for the elite Ura, though I didn't have all 10 idols on when I played that one.

This last one was really hard for me with all 10 idols turned on, but I found that the Calamity Cannon to be very useful. I upgraded the Cannon so it would have speed reload and speed charge, as well as the final upgrade that makes the missiles home in slightly, and I took the tonic that boosts your damage by 15% but makes you lose a continue, along with the life leech, health boost, defense boost, and a few other tonics I forget. The Cannon has enough range to home onto the turrets and stinkeye spawners when they're out of your vision, making it very easy and safe to take care of them. And as far as the Ura go, the Cannon doesn't exactly destroy them, but it is still very effective. Since the melee Ura sort of 'phase' towards you, that usually give you enough time to quickly fire off one or two rockets before they close the distance, knocking them back and leeching you a ton of life. For the Ura I found the best strategy to generally be to try to take care of any ranged attackers first with a few cannon shots then just rolling towards them to take them out with the spear really quickly while not engaging any melee Ura until all the other threats were taken care of, then dealing with the melee Ura by falling into a 'block, hit, block, hit' pattern with my spear. And when there were multiple melee attackers, I used the Brusher's Sweep skill to push them away to kill one with the cannon and maybe push the others off the edge, or I suppose you could use the Stealth skill to set up a critical rocket, hopefully killing all of the attackers.
 
So while I found the cannon to be useful personally, I don't think it would pair very well with the mortar since they both have the same weakness (melee attackers). I managed to do it with the spear and cannon, though I think that the hammer and cannon might be better at handling the melee Ura since the hammer deals more damage per hit, and you generally won't want to try to get in more than one attack before you get your shield back up since the Ura attack very quickly. Good luck with the challenge, it was by far the most difficult for me.

The Mortar slaughters everything.

That is all.