Kobolds are just worse deep elves
Mage backgrounds
I do not think I need to explain why DE is better at these. Although Ko gets faster XLs and a better fighting aptitude, the huge differences in magic aptitudes and intelligence mean that the DE can actually get more fighting skill.
Berserker, Abyssal Knight, Chaos Knight, Skald, Fighter, Gladiator, Monk
Going just by aptitudes, Ko would look to be better than DE at these. However, there's a big problem: these backgrounds need to use melee weapons, Ko is small, and small species aren't allowed to wield many weapons. Picked maces? Have fun not being able to wield dire flails (the best common mace) or great maces. Long blades? Your aptitude is actually worse than DE's and you can't wield great swords. Axes or polearms, you might as well have chosen unarmed. Speaking of unarmed, guess which species has an easier time casting the form spells that make unarmed stop sucking?
Now of course, if you went with short blades or staves, your size doesn't matter...but if you went with short blades, that's its own punishment. And furthermore...
All of the above plus the launcher backgrounds
You know what's really good? Ranged combat. You know what DE has a +1 aptitude in and kobolds can't even wield? Longbows. Kobolds can't even use javelins. Pretty much every non-Mage background wants to throw or shoot stuff, and DE outperforms Ko at throwing stuff (because it's actually allowed to throw javelins) and at shooting stuff (because it's actually allowed to wield the best launcher).
Enchanter
Okay, well, Ko at least has Enchanter, right?
Ko has +2 dodging, +0 spellcasting, +0 hexes, +4 stealth. KoEn has 5 str, 16 int, and 15 dex.
DE has +2 dodging, +3 spellcasting, +3 hexes, +3 stealth. DEEn has 5 str, 19 int, and 15 dex.
Hey you know what really helps Enchanter? Hexes.
Why this is a problem
So Ko is better than DE at using inferior weapon classes and at swapping vampiric weapons (carnivore). DE is better at pretty much everything else. This is not a good look. It's not a huge disaster that DE is weaker than Hu, because there are still certain things DE is much better at (casting spells). But when one species is worse than another at pretty much everything, why play the species at all?
Possible solutions
Remove kobolds
This has come up many times for its own reasons, and I have to include it because of that. Removing kobolds would indeed fix the problem. It usually comes up in relation to halflings being redundant, but I think Ko is more redundant with DE (as you might have guessed from the rest of this post).
Buff kobolds' HP
Ko has better Fighting and XL apts than DE, but ends up with little to no actual HP advantage on spellcasting characters, and its HP advantage on Berserker is still not very big. Changing their crippling -2 HP aptitude to 0 (or possibly even better) would at least give them better defenses than deep elves.
Let small species use the same weapons as medium species
DE is better than Ko at using weapons because Ko isn't allowed to wield the best weapons, and some other weapons (such as tridents) become two-handed for them. Removing small species' weapon restrictions would solve this problem and make Ko decisively better at melee.
Buff kobolds' offensive power in some other way
If you want to keep the weapon restrictions, you could still make Ko more competitive by giving them huge aptitudes in those weapon classes. A two-handed broad axe at an aptitude of -2 sucks, but a two-handed broad axe at an aptitude of +4 might not. Tomahawks at +1 throwing compare really badly to javelins at +0 throwing, but tomahawks at +5 throwing might not.
Nerf deep elves' non-magic stuff
You can go the other way too, and give deep elves like -4 aptitudes in all weapon classes and stealth. I do not particularly like this as it makes DE even more one-dimensional, but it would make Ko better than DE at some things while avoiding any possible power creep.
Remove deep elves
I mean, I don't think anyone actually wants to do this, but it would, technically, solve the problem just as much as removing Ko would.