The court said IEBC, which is the first respondent, should make arrangements for the read-only access to be conducted in the presence of the deputy registrar, political parties and their agents within five days.
However, the court declined two applications by Mr Lesirma.
He had asked the judge to order IEBC to give him full and unfettered access to every biometric electronic appliance used at poll stations.
He had also asked the court to order the commission to provide access to all parties, for scrutiny and supply to the court of original forms 37A, B and C.
Last month, an application by the Samburu County government seeking orders to strike out the petition for failing to enjoin the deputy governor was dismissed by the court.
A plea seeking an order for striking out and expunging from court records respondents’ answers for want of service within the statutorily prescribed time was also thrown out.
However, the court allowed an application by Mr Lesirma to file additional supplementary affidavits to support his case.
The case will be mentioned on December 18.
In a related development, former Turkana Senator John Munyes, who is challenging the re-election of Governor Josephat Nanok, now wants a scrutiny of all materials used in the General Election.
Through his lawyer Katwa Kigen, Mr Munyes was given a nod by the court to make an application for the scrutiny to be heard on December 14.
This is after the court completed the hearing of the petition on Tuesday.
Mr Joshua Ang’elei, the lawyer who has teamed up with Mr Kigen, told reporters that the scrutiny would require access to election materials like polling day diaries, ballot papers and Kiems gadgets.
Mr Munyes says the election was marred by malpractices and that Mr Nanok was involved in bribery.