How do you defeat the arch mage and the undead night?

My suggestion would be to take out whoever has the most stars first, if the stars are equal nail the Mages right away, they shield, heal, and do a bit more punch in their damage, but tend to be a bit more squishy to damage, save your mana to 100% before delivering a finishing blow. Make sure to stock up on potions, mission satchels are a good way to build up a number of them.
If worse comes to worse and you just get too stuck level up your pets a bit more and maybe take a look here for a build guide. I personally use a Physical damage tank, and then a magic damage dealer and a physical damage dealer for my team. Any more questions and I would be happy to help just shoot me a ping or PM :)

Whenever I kill them, they just come back again.

How many have you killed? It's quite possible you just ran into a hoard and there could be several replacements to take the spot of the one you killed. It's been a while since I did the quest, but that's what it sounds like.

I am having the same issue as well in Enigmatic Grotto.
I had a team with a tank, battler and magic hitter and all were near the same level as the arch mage but herein lay a problem for me.

Having a team with only one battler makes the team vulnerable and while the guides are really great, it is just not working for me.

See..having a tank is really great as it draws attention away from the others however it is also pretty useless if I have only one battler to do physical stuff and one to do magic damage. It is not feasible to use all points on one for, say, intelligence and agility while still being basically a useless battler then have another who has the strength but little else and a tank that has health and agility but nothing else...to me this makes a team very lopsided.

So what I am trying out is to make both my tank and battling pets useful enough so I have two heavy hitters to take out the tougher enemies, having two fighters cuts down on the battle duration. I am not so sure what good magic damage will do against the undead especially if you cannot cast spells per se.

Another issue is the potions, not only are they pricey but also that you use them at the same time an enemy attacks so the difference between recovery rate and damage taken at a single turn is negligible.

Having to wait 25 turns before being able to utilize "mend" is also a sticking point although it also seems to recover more health at one turn than the potions you may have available to you at that point.

So I want to get through the Enigmatic Grotto but, with all due respect, following the guide has not helped me much so I am going to experiment and develop my tank pet into an effective battler as well then my magic damage pet can be the ninja so-to-speak and maybe deal damage in a covert way....or something.

Either way the arch mage is a pain in the spleen.