As you had said:
2) however, the letter of offer that I signed explicitly states that "the letter of offer is only effective after both parties sign on the letter of offer"

That's the last nail in the coffin for your case. I don't think you can go any further than that. Oral representations/terms can't replace terms put in writing. Section 91, Evidence Act 1950. Sorry, mate. 

Don't get me wrong, mate. I didn't say that oral contracts are unforceable. Oral contracts are 100% enforceable, no doubt. So are contracts by way of conduct. There is nothing which says that a contract must be put down in writing.

What I stated refers to the gist of your case - i.e. that since there already are terms put down in writing (as per the letter of offer), then any oral evidence to the contrary cannot be adduced and accepted which contradicts those terms.

Let me illustrate:

Vendor's case - terms of the letter of offer states that "the letter of offer is only effective after both parties sign on the letter of offer". The purchaser had signed the letter of offer. I did not sign the letter of offer. As such, the letter of offer does not bind me.

* As you can see, he is relying on the terms of the letter of offer.

Your case (as purchaser) - you signed the letter of offer, you paid the 2% cheque and the owner verbally told you that he is agreeable. The agent also sent you and email confirming same (I assume so as you did not delve into its contents).

* You, on the contrary, is relying on the vendor's oral agreement, and the email by the agent.

Before the Court, you cannot adduce evidence of the vendor's oral agreement nor the contents of the email by the agent to support your claim that the vendor has accepted your offer to purchase the property. Why? Because those evidence contradicts the written terms of the letter of offer which clearly stipulates that "the letter of offer is only effective after both parties sign on the letter of offer".

So, did both parties sign on the letter of offer? No.

End of story.

Hope the explanation above suffices for your understanding. Sorry if it's not clear enough. If you're still not satisfied, you can always seek professional legal advice.

Good luck 

PS: Common law does not apply when there is statute. Common law cannot supersede statute, i.e. the Evidence Act in this case.