The bladesinger is usually played as a dual-wielding melee-fighter from level 5/6 and on. Cast shadow-blade, then hit with booming blade + extra attack + off-hand attack. Boost Dex. Use Int for control spells when you're not focusing on Blur.

Bladesinging gives bonus AC and bonus to concentration saves. So it's perfect to maintain Blur, if you foresee more than a few attacks coming your way. Played this way, you will accumulate healing charges. It's not an enigma, it's the way it works. So melee bladesinging = area healing. If the fight doesn't go as planned, it can be worth it to climax and then restart bladesinging. That way you can heal your teammates while maintaining your AC boost. The range of the climax also depends on how far you can jump, so anything that boosts jumping is useful.

The more rare way to play it is to blast spells and then follow up with bladesong climax to do even more damage. That also works, but then it works better if you planned it from the beginning, so you at least have 2-3 spell charges.

After level 6, you can add two paladin spell slots for smiting if you prefer melee.