I don't know enough to comment on the speeding ticket, but you admitted to running the stop sign. Like cw said, get a lawyer, cop a guilty plea to running the stop sign, and see if your lawyer can get the State to drop the speeding charge in exchange for the guilty plea on running the stop sign.
And just some additional..... when the cop said he "estimated" your speed, he probably paced you the 2 blocks after running the stop sign before he activated his emergency lights. A pace has been deemed an acceptable way of obtaining a vehicles speed by the US Supreme Court. A pace is performed when a cop sets a speed either in front, behind, or beside your vehicle and maintains a constant speed with his/her speedometer and, during which time, the officer neither loses or gains any distance between his/her and your vehicle. Unless you know, for a fact, how fast you were going, you are more than likely going to be found guilty if you plead not guilty to the charge unless you can prove to the court, beyond a reasonable doubt, that the officer who issued the citation was mistaken.
Then again, you could go to court and, if the cop that issued you the ticket isn't in court, plead not guilty and the ticket will be thrown out because the cop won't be there to testify for the State.