As Akali you are a queen of both burst and sustained damage, having abilities on short cooldowns with no mana pool to worry about. Morgana is not a counter for you and you should be able to defeat her if you play to your abilities and against hers.
First, Morgana's damage is easy to avoid. Her snare is slow moving and easy to dodge, especially if you are around your minions, and her tormented soil is a very small AoE that you can step around. Even without boots you shouldn't be taking much damage from Morgana unless you are engaging.
If you are taking damage, do not forget that you yourself have spell vamp! Morgana has 15% early game while you have 10% from your passive if you take an item giving you 10 attack damage. You should really start with a Doran's Blade as Akali, which is a combined 10% spell vamp and 3% lifesteal. Even better, select runes that give you spell vamp without having to buy items, then start with boots in order to dodge Morgana's skillshot. Morgana's abilities have 10 second cooldowns each, and yours are only 7 seconds decreasing per level, meaning you have more opportunities to vamp. The only edge Morgana has on you is that Tormented Soil has a large life return when used on groups of stationary minions; make sure you get a similar return (although it is much less) from Crescent Slash.
Don't get zoned! Morgana can't do much to you if you are in a minion wave except auto-attack. This will get your minions attacking her, and you can follow up with them for more damage. If you let Morgana push you out of your minion wave you'll have a hard time getting back unless you have your ult up.
Because Morgana's abilities have a large cooldown, take action when she uses one. In particular, watch for her using Dark Binding and Black Shield. If she has either down it is your opportunity to get in there. When Dark Binding is the only spell down Morgana's Black Shield does not make her immune to damage; it breaks after enough damage and your burst should be able to break it. Keep in mind that your abilities reset auto attacks, so alternate abilities and auto attacks to maximize damage while her shield is up. If her shield is down but Dark Binding is up, you might have trouble engaging if the Morgana is exceptional with it. Try to bait it out of her if possible.
Keep in mind that it's not always about killing your opponent. If the Morgana you are against saves her Dark Binding and Black Shield and is always using them at the right time, give up trying to kill her and focus on farming. Akali is amazing at farming, being able to reset auto-attacks and having two great damage spells. You may not outfarm Morgana given how awesome Tormented Soil is, but you will come close. Don't forget that you can return that pressure with lots of auto-attacks and using your abilities on minions - if you really feel like you cannot kill her you should always push back on her minion wave as hard as she pushes yours. If the Morgana pushes too hard signal to teammates (especially your jungler) and she will be easy to gank.
In most cases you can at least hold mid as Akali against Morgana. Both of you have good late game potential as well, so don't give up. If a teammate could do a better job than you in mid then let them do it. If you have to be mid just focus on farming, try your best, and don't get psyched out. You will get better and better as Akali and Morgana will only mid in less than 5% of the games you play anyways, she's not a typical mid nowadays in League of Legends.
