Then you have a good base for taking them out. Warlord lends itself well toward armies of normal units. Halflings and High Men both have decent units. A large stack of halfling swordsmen is sufficient to kill one AE, but the stack will take a lot of losses in the process, so it isn't viable against a larger force of AE.

You've seen 4 Air Elementals. If there are 5 other units in the node, then you know there can only be 4 AE. Look at the enemy formation; if there are odd holes in it, an AE probably occupies that slot.

Be aware that all combat spell solutions will be subject to the Node's dispelling effect. Sometimes a combat spell will work, but don't depend on them working because the node will catch most of them. To be sure a spell succeeds, either it must be of the same school as the node (sorcery nodes allow sorcery spells) or you must have the Node Mastery retort.

Nature tools
-Web can pull an AE from the sky, allowing your melee units to start initiating attacks on it for the rest of the battle. The web will also delay the AE from attacking, but probably only for one turn while it breaks the web. That said, the node will probably block this spell from working.
-Several Basilisks should be able to get the job done. Their stone gaze should let them initiate on the AE even if the AE are immune to the stoning effect. Basilisks have low defense, but they have very good attack and hp values.
-Giant Strength is pretty cheap in both casting cost and maintenance. Adding it to some of your attackers may help tip the scales in your favor. Make sure to cast it before the battle, because the node will block it.
-Stone Giants should be able to get the job done, but you'll probably want at least one SG per AE, and you should expect to lose a few.
-Gorgons are strong enough to get the job done, and they fly so they can initiate fighting. You might need two units of Gorgons to kill all the elementals.
-I'd wager that a single Earth Elemental will get the job done for all 4 AE, though you might need something to finish off the fourth one. However, the node might block the EE spell. Sadly, the EE has to wait for the AE to target it.
-Iron Skin on a strong attacker (cast before the battle) should provide a lot of protection against the AE attacks.

Death tools
-Most Death creatures have Immunity to Illusions, so bringing even a single unit of Skeletons will let you know exactly how many AE are in the node. Skeletons and Ghouls can be good for cleaning up any Phantom Warriors that are in the node with the AE, but both units are going to be ineffective against the AE themselves.
-A large stack of Werewolves might be able to cut it, but I'm not sure. On the bright side, if you win then all your dead werewolves come back to life, and they can also clean up Phantom Warriors without much issue if your werewolves aren't too badly damaged.
-Several units of Shadow Demons might be able to get the job done, and they'll be able to use their ranged attacks against the AEs too! SDs have somewhat low stats, so several SDs will probably go down in the fighting, but Regeneration will bring them back if you win.
-A few units of Wraiths will probably get the job done, but there are better uses for any Wraiths you control (like taking over cities defended by normal units).
-Black Sleep can let anything able to hit a flyer murder an AE, but between the node dispel effect and the AE's very high resistance, this is more theoretical than a viable tactic.

Halfling tools:
-Build an Alchemist Guild and a Fighter's Guild, then you can spam Slingers. A stack of 9 slingers should be able to handle the node, and they can tear apart any Phantom Warriors or Beasts accompanying the AEs while waiting for the AEs to die on them.

High Men tools:
-Build an Alchemist Guild and a Fighter's Guild, and spam a bunch of Pikemen. They'll get the job done. Take about 1.5 pikes per AE and you'll be more than fine.
-Paladins will get the job done, just like they do against pretty much everything else.

TL;DR
When balancing cost versus effectiveness for this scenario, my top picks are Pikemen and Basilisks (or a mix of both, allowing you to put cities and magic to work at the same time). Slingers and Shadow Demons are other good choices that can be added to the stack. Shadow Demons are perhaps the worst of these four options because they are rather expensive but slow; Pikes and Slingers are both quite cheap for what they do, and Basilisks are expensive but faster. However, Shadow Demons can fly, plane shift at will, and regenerate, so a large stack of them are pretty good at reaching anywhere (eventually) and wrecking havoc on arrival.