Hit Dice: d8 or d10.
D8 for certain.

Skills: Choose two
Need a list to choose from.

Spellcasting
A Martial fullcaster, huh? That's going to be a bit difficult to pull off. Closest things to a martial fullcaster are the Druid, Bard, and Pact of Blade Warlock. Druids perform too specifically martially, through Wildshape and Shillelagh, so we can put the Druid aside. That leaves us with the Bard and PoB Warlock. Both are decent martials, but hardly compare to the martial might of the Fighter, Rogue, Monk, and Barbarian. What they lack in martial ability, they easily make up for through the full range of casting.

So if you want to make your Gish a fullcaster, you're going to have to tone down its martial prowess.

The classes the achieve the fullest balance between martial ability and casting are the Paladin and the Ranger, both being Halfcasters. I might recommend you modeling it after them, rather than the Bard.

Steel Focus

As a battlemage you have the ability to create a bond between yourself and a weapon. Establishing the bond can be done over a short rest. You know the location of a weapon bonded to you as long as it is on the same plane, and can use it as an arcane focus. You can be bonded to only one weapon at a time. Bonding a second weapon breaks your current bond, or fails.

Fighting Style

As the fighter ability, but you can take only Defence, Dueling or Great Weapon Fighting.
Considering the presence of Dueling, I would have expected your Battlemage to have proficiency in shields. But it doesn't. Wizards don't typically use shields, but I can imagine a Battlemage using shields prolifically.

Sword and Sorcery

At second level, your bond with your weapon allows you to make openings for your blade with your spells, and vice versa.

On a successful attack with your bonded weapon against an enemy that you successfully attacked with a spell or that failed a save against your spell last turn, you deal extra damage equal to your intelligence modifier plus the level of the spell.

You have advantage on spell attack rolls against enemies you damaged with your bonded weapon last turn. Such enemies also have disadvantage on saves against your spells.
The latter feature is too powerful. It grants you unlimited on-demand advantage/disadvantage with your spells at 2nd level. The Eldritch Knight's 10th level feature only grants disadvantage on a creature's next save against the EK's spells. You've taken the EK's 10th level feature and then essentially double its power, and this isn't including the other feature at 2nd level. I recommend leaving the former spell as is and moving the latter feature to a much higher level, perhaps around 14th.

That's all I have so far. It needs a capstone (something to regain spells, judging by the other casting classes) and subclasses. I'm thinking there will be two subclasses, an offensive one (which I'm currently calling either Spellsword or Arcanamach) and one based on protecting the party against incoming spells (Abjurant Champion). I don't know what abilities they should grant though. I don't want to just copy the abjurer's abilities. The spell list will be a cut down wizard list, mostly losing single target enchantments (charm person and similar), buffs (because they don't work well with Sword and Sorcery) and the really esoteric stuff (feign death, demiplane).
Capstone could be something related to action economy. This class is meant to meld Sword and Spell together, right? Maybe at 20th level you can cast cantrips that have a casting time of 1 action using 1 bonus action instead.

Have you seen the Guide to Homebrewing Classes?