I've tested Eomer with the new patch and although he has not been changed, the rest of Rohan being tweaked hides some of his weaknesses.
But he still is not as Tolkien introduced him to us through his books.
So here are my suggestions once more:
Lvl 1: Mount/Dismount -> as it is. Maybe change the name to Firefoot?
Lvl 1: Piercing Spear -> Instead of targeting a unit the spear should be thrown to the ground.
It can fly a great distance and pierces any unit it comes across killing it and throwing the surrounding units in the air.
When it hits a monster unit the spear will stop and cut the monster's health in half.
When the spear hits a hero the damage done to him will be moderate but it will stun all allied units to the hero in a wide radius.
These effects will become more powerfull according to Eomer's levels. (3,6,9 respectively)
Lastly the spear's speed should be greater.
Lvl 3: Leader of the Exiles -> as it is.
Lvl 5: Memorial -> Although it's a nice adaptation from the movies and it has some usefulness it has a small range (in order to be activated and it's too slow). This ability should either be changed in order to happen more quickly, or changed as
Lvl 6: Gúthwinë (active) -> Eomer's the greatest fighter in Middle- Earth second only to Aragorn, he wields the great sword Gúthwinë, in its sight the enemy units cower and lose all leadership bonuses and 30% armor and damage, while Eomer recieves those amounts to his own stats.
Thus the supporting role of Eomer still stands, and the lore is followed more strictly showing what his abilities actually were.
Lvl 10: King's Cold Fury: Eomer leads his troops to battle waving his banner through the air, all allied units in a wide radius gain attack damage, recieve a great amount of armor, while Rohan's cavalry gain trample damage and moderate protection from spears.
I think that is what Tolkien had in mind for Eomer, and these suggestions will aid Rohan to have a great hero, whom it lacks very badly and thus to get a way to turn the tide to its favor.
Let's make this possible people!!