As for the whole chords and cords thing...

Sources:
http://www.Dictionary.com
http://www.en.wikipedia.org


Chord-
1. The string of a musical instrument.
2. A combination of three or more pitches sounded simultaneously.
2a- A combination of three or more notes that blend harmoniously when sounded together

3. A feeling or emotion.
4. The line segment between two points on a given curve.
5. To establish or play a chord or chords for (a particular harmony or song); harmonize or voice: How would you chord that in B flat?
6. The string of a musical instrument.
7. Bring into consonance, harmony, or accord while making music or singing.

Cord-
1. A long rope-like structure, such as a nerve or tendon. ie: A spinal cord.
2. (a piece of) thin rope or thick string
3. A small, flexible, insulated cable.
4. A slender length of flexible material usually made of twisted strands or fibers and used to bind, tie, connect, or support.
5. A string-like part of the body.

And the sentence is: "Finally my chords of steel is starting."
Though should it be "are starting" rather than "is starting" because "chords" is plural?...

Chords of steel makes sense AND cords of steel makes sense. It depends what he's talking about.
"The string in his musical is instrument is steel", makes sense, he could play like the violin or something, and have the CHORDS on the violin be made of steel. But in this case, he means his vocal CORDS. But his VOICE can also be a "musical instrument" in a way. And sound is generated in the larynx, and that is where pitch and volume are manipulated. Now if they had made Apollo's phrase "Larynges of steel" it would make good sense except for the part where it says "Larynges" because we one have one larynx(I think). And who said he was talking about his vocal folds? He cord be talking about...his spinal cords...

Spinal Cord theory:
Some people happen to break their spinal cords, and sometimes, it's not fixable. Either they're paralyzed from there down, or they have steel rods put in/on/around(i'm no doctor) their spine so that they could move it. STEEL CORDS(since cords also can mean "rope" as in rods in his back) DOWN YOU SPINAL CORDS. "Cord" and "steel" in one sentence! Cool! *ahem* Anyway, if we knew more about his past, it could be that sometime in his life, he broke his spine. And they had to have steel rods put in his spine/back. And so then he starts exercising and he says "Finally my cords of steel is starting". As in the cords down his back are starting to work and he could move. But in this case, he says CHORDS instead of CORDS.
But of course having "chords" down your spinal cord also makes sense but...violin strings down your back?...umm..what the hell?

So he can't mean his spinal cord.
Observe:


He can also mean his ventricular folds.
Ventricular Fold-
Either of the upper two vocal cords that are not involved in vocalization.
The ventricular fold is right above the vocal cord and is also cord-like. And of course the vestibular fold.

The vestibular fold (ventricular fold, superior or false vocal cord) is one of two thick folds of mucous membrane, each enclosing a narrow band of fibrous tissue, the ventricular ligament which is attached in front to the angle of the thyroid cartilage immediately below the attachment of the epiglottis, and behind to the antero-lateral surface of the arytenoid cartilage, a short distance above the vocal process.

Observe:


He could be talking about that too. His "false" vocal cords making a chord.

But the most obvious answer here is that he's talking about his vocal CORDS making a CHORD. The vocal cords, make the "Objection!" and the his CHORDS of steel meaning his PITCH and VOLUME of the "Objection!".

Also, his CHORDS of STEEL. His "Objection!" sound is like steel. Using his vocal CORDS. His PITCH of steel.
His vocal CORDS play a CHORD of CHORDS. Chords are has something to do with music, right? So He's using his vocal cords, to vocalize a chord.

Take a look at Chord definition #7.
"7. Bring into consonance, harmony, or accord while making music or singing."

Singing. He's singing with vocal cords of steel yet vocalizing steel-made chords, or pitches.
Using vocal CORDS of steel(a metaphor, nobody has vocal cords that are steel) to vocalize combinations of tones or CHORDS that are referring to the musical STRINGS in his voice box or larynx that are made of steal(also another metaphor, because I believe nobody has steel strings in thier mouth, unless they have recently swallowed some an are on their way to the hospital).

Here's what is causing some confusion:

Vocal cords= string-like
Chords are the strings on a musical instrument and in this case the musical instrument is Apollo's
vocal CORDS.

Cords. String in someone's throat. They're the same thing. When they say "Chords of Steel" they mean the STRINGS IN APOLLO'S VOICE BOX(larynx) ARE MADE OF STEEL. Because Apollo's voice box is an instrument.

Here's how that works:
Musical Instrument- Apollo's Voice
Which Chord- The strings in his voice box, AKA the Vocal CORDS
The strings can ALSO be called cords.
Because a cord is also a string, but a chord is also a string.
So his vocal cords can be called his chords because his chords are the strings in his voice.

They should have put "cords" to make it more clear.
Observe:

"Finally my cords of steel are starting"
His vocal cords of steel are working

"Finally my chords of steel are starting"
The strings in his voice box are starting to work.

Either way they're talking about his vocal cords being made of steel. But if you say "chords" instead of "cords", then it can also mean that his voice sounds like steel, though steel itself has no sound. When his voice sounds like steel I'm guessing it means it has some relation to the properties of steel.
ie: Steel is strong. His voice is like steel, because his voice is strong.

The strings in his voice box can mean his vocal cords OR some other kind of cord, but I think it's pretty obvious what they are talking about.

Very confusing.

To put it simply, it was probably meant to be a metaphoric-pun.