Why Didn't Harry Use The Mirror?

Two, related reasons.

He doesn't know what it actually is

Sirius is vague when he gives the mirror to Harry, and it's clear that Harry never opens the package until it's too late:

'I want you to take this,' [Sirius] said quietly, thrusting a badly wrapped package roughly the size of a paperback book into Harry's hands.

'What is it?' Harry asked.

'A way of letting me know if Snape's giving you a hard time. No, don't open it in here!' said Sirius, with a wary look at Mrs. Weasley, who was trying to persuade the twins to wear hand-knitted mittens. 'I doubt Molly would approve - but I want you to use it if you need me, all right?'

'OK,' said Harry, stowing the package away in the inside pocket of his jacket, but he knew he would never use whatever it was.

^{_{Harry Potter and the Order of the Phoenix Chapter 24: "Occlumency"}}

As far as Harry knows, it's some sort of emergency beacon; he doesn't know that it's a general means of communication, and he doesn't care to know.

He'd forgotten about it.

The context in which Harry learns what Sirius gave him makes it clear that he didn't often think about the gift:

Harry pulled some crumpled robes out of the very bottom of his trunk to make way for folded ones and, as he did so, noticed a badly wrapped package lying in a corner of it. He could not think what it was doing there. He bent down, pulled it out from underneath his trainers and examined it.

He realized what it was within seconds. Sirius had given it to him just inside the front door of number twelve Grimmauld Place.

^{_{Harry Potter and the Order of the Phoenix Chapter 38: "The Second War Begins"}}

Harry's been having a pretty rough go of it through OotP, and especially in the hours leading up to needing to contact Sirius. It's perfectly understandable for this unassuming package, which he explicitly told himself not to use, to have slipped his mind.

The Blue Eyes Thing

To correct your misconception, Harry sees nothing when he tries to use the mirror except for his own reflection:

[Harry] looked around to make sure there was nobody else there; the dormitory was quite empty. He looked back at the mirror, raised it in front of his face with trembling hands and said, loudly and clearly, 'Sirius.'

His breath misted the surface of the glass. He held the mirror even closer, excitement flooding through him, but the eyes blinking back at him through the fog were definitely his own.

He wiped the mirror clear again and said, so that every syllable rang clearly through the room:

'Sirius Black!'

Nothing happened. The frustrated face looking back out of the mirror was still, definitely, his own.

^{_{Harry Potter and the Order of the Phoenix Chapter 38: "The Second War Begins"}}

The blue eyes you remember seeing may be a reference to the movie, since Daniel Radcliffe (in)famously has blue eyes, but it may also be a reference to Deathly Hallows, when Harry periodically stares into a shard of the mirror and occasionally sees Aberforth Dumbledore peering back at him.