What is unbalanced about this homebrew College of the Storm bard subclass?

Question

What, if anything, is unbalanced about this homebrew College of the Storm bard subclass?

I think the subclass I made is unbalanced because of the number of abilities gained at third level, and the combo moves you can pull off with Stormy Blades plus something like thunderous smite - but I am not sure how to balance it while keeping the feel of this subclass.

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The subclass:

College of the Storm

Bards of the College of the Storm use the creative magic of thunder and lightning. They bask in the beauty of the storm.

Bonus Proficiencies

You gain proficiency with medium armor.

Stormy Blades

When you join the College of Storm at 3rd level, you gain the ability to make your weapon attacks magically imbued with thunderous energy.

Once per short rest when you hit a creature with a weapon attack, you can choose to deal an extra 1d6 lightning damage to that target. This damage increases when you reach certain levels in this class, increasing to 2d6 at 5th level, 4d6 at 10th level, and 7d6 at 15th level.

Bonus Spells

In addition, when you join this college at third level, you learn the thaumaturgy cantrip, and one additional spell from any class that deals lightning or thunder damage the spell must be of a level you can cast. When you choose this spell, it becomes a bard spell for you. If you choose to replace this spell when you gain bard levels, you must replace it with a spell that deals lightning or thunder damage.

Extra Attack

At 6th level, you may attack twice, instead of once, whenever you take the Attack action on your turn.

Storm’s Fury

Starting at 14th level, when you are hit by a melee attack, you can use your reaction to deal lightning damage to the attacker. The damage equals your bard level. The attacker must also make a Strength saving throw against your spell save DC. On a failed save, the attacker is pushed in a straight line up to 20 feet away from you.

Answer 1

One weird thing about your subclass compared to other bard subclasses:

Nothing interacts with Bardic Inspiration. Consider changing that.

Level 3 - proficiencies and bonus spells are about on par with the other melee bard subclasses.
Notably though, Stormy Blades is strictly weaker than Psychic Blades from the College of Whispers.

Consider powering Stormy Blades with Bardic Inspiration, and giving it something to differentiate it from Psychic Blades - like a small AoE effect, an obscuring thundercloud when you use it, or minor flight on use like the Storm Sorcerer gets for casting spells.

Level 6 - Nothing new here.

Level 14 - Other melee bard subclasses get an always-on ability that adds an average of 7 or 9.5 damage per turn used (for Swords or Valor).
Yours adds 20 damage per turn, plus an additional reliable (STR save is one of the uncommon saves) effect.
1/2 level damage would be an appropriate change in terms of power, as it would have higher damage and reliability than the others, but is putting the user in more danger.

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A note about subclass features and levels:
The level 14 subclass feature of one class (for example, Sorcerer) should only be used as a power comparison with other subclasses of the same class.
This is mostly because the base classes hold different amounts of power that is made up for by subclasses.

Case in point, the base Sorcerer class is sparse - you only get the core features of spellcasting and metamagic. Most of the interesting and powerful features come from the subclass.

Conversely, the base Bard class is loaded - in addition to the core features of spellcasting and bardic inspiration, you also get Expertise, Jack of All Trades, and Magical Secrets. As such bard subclasses are, as a whole, weaker than Sorcerer subclasses.

Answer 2

I'll do a balance breakdown and give some suggestions, by comparing this to the closest sub-classes currently available to the Bard: the College of Valor and the College of Swords.

level 3

All three grant Medium Armor proficiency. Valor also grants martial weapons and shields, which is better and gives more options. So that's fine.

All three also grant an ability that adds additional damage to attacks. Out of the three, yours is the only one on a "once per short rest" timer, the other two use Bardic Inspiration dice. Yours scales in damage directly. If anything, for consistency, I'd make yours add extra Thunder and/or Lightning damage through Bardic Inspiration as well. That seems to fit a bit more with the Bard theme.

Since the other two both have an additional ability (the option to boost AC with Inspiration and a Fighting Style respectively), adding on a cantrip and a spell known seems fine. It might even be a bit on the weak side at this level.

level 6

All three sub-classes grant Extra Attack at this level. Nothing to see here, moving on.

level 14

This ability does considerable guaranteed damage, and appears to have no restriction at all. You could potentially use this every turn as long as you're in melee (where you belong), which makes it very powerful compared to the 14th level abilities of the other two sub-classes.

It's likely to deal a lot more damage than the Valor Bard's Battle Magic ability (which uses Bonus actions instead of Reactions) and almost certainly more than the Swords Bard's 1d6 extra damage per swing that doesn't use an Inspiration die.

The ability seems to fit the theme just fine, so I'd suggest putting some kind of cap on it. Either make it use Inspiration (which makes it a powerful inspiration ability, but that's probably okay at this level), make it once per Short Rest, or turn it into a special spell ability (a Lightning version of Hellish Rebuke) so it uses up spell slots.