The Senate in October voted to remove Rigathi Gachagua, the previous deputy president, from office in October on charges including corruption and stirring ethnic hatred - accusations he rejects and has dismissed as politically motivated.
Kenya's High Court had put Kindiki's inauguration on hold after Gachagua and his supporters filed more than 30 court challenges to prevent his dismissal and replacement.
But the stay order was lifted on Thursday, with judges saying that the deputy president position should not remain vacant according to the constitution.
Gachagua's legal battles extended months of political turbulence in the country, East Africa's largest economy, that began with nationwide protests against unpopular tax hikes in June.
President William Ruto invited opposition members into his cabinet to create what he called a unity government in July in an attempt to ease tensions after demonstrators stormed parliament.
The political realignment also sidelined Gachagua, who during Ruto's election campaign helped secure a large block of votes from the populous Mount Kenya region. The two men have since fallen out.
Ruto has not commented on the impeachment proceedings but in October called for national cohesion and an end to "tribalism and exclusion".
Gachagua has compared the government to a shareholding company, suggesting that those who had voted for the ruling coalition deserved preferential access to government jobs.
Speaking after Kindiki's swearing-in, Ruto told his new deputy to "serve the people of Kenya equally, those who voted for us and those who did not vote for us. The government of Kenya equally belongs to all the people."
Kindiki was a leading contender to be Ruto's running mate during the 2022 election and was appointed interior minister shortly after the president took office in September that year.
