Let's quote the goddamn Takings Clause of the Fifth Amendment:
...nor shall private property be taken for public use, without just compensation.
That's as clear as any constitutional provision can be. The petitioner's argument was, essentially, that the Clause absolutely prohibits the government from taking private property for private use. A person who can speak English and follow simple logic cannot possibly reach such a conclusion.
First, the Takings Clause says nothing whatsoever about taking for private use. It says that, if the government wants to take your property for public use, it has to pay you "just compensation". It can take your property, it just has to pay you. (Whether "just compensation" is fair market price or something else, is a legitimate debate issue.) But it says nothing at all about "private use", and it is simply idiotic to conclude that anything in the Clause implies that such takings are forbidden.
Second, what would it even mean for the government to take property for private use? That's a contradiction in terms. The government is a public institution, and whenever it takes property, it is for public use. That doesn't preclude some private use: governments do sell (or give away) property to private individuals or businesses, just like they often outsource provisions of public services to private industry. But a government cannot, by definition, do anything that has no public purpose.
Suppose the local government takes your land (and your house on it), pays you just compensation (and assume, for the sake of argument, that everyone involved agrees that the compensation is just), holds it for 100 years as the idyllic Fifth Amendment Park, and then sells it to a private business that wants to build a shopping mall on it. Would this violate the Constitution? Would any of the dissenters in Kelo even think that it might violate the Constitution? If it did, then every sale of government property to private entities is unconstitutional. You cannot draw a distinction based on how the government acquired the property, because that would in effect give the original owner continued property rights - in other words, the taking would never be complete. But that would be absurd; not even the staunchest libertarian would argue that the government doesn't really own the property for which it paid full price.
Now change only one thing in the previous hypothetical example: instead of 100 years, the government holds the property for one minute. Can this change the constitutionality of the described actions? I challenge anyone to find anything in the Constitution - and feel free to search the penumbras as well - that would make this time interval relevant. There is nothing, and the example now resembles closely what happened to Susette Kelo. This is a no-brainer: no constitutional violation occurred. (There could have been a dispute over whether the compensation was just, but that was not the issue brought before the Court.)
What the city of New London did may well have been bad policy. The municipal and state laws that allowed it may well have been bad laws. While the Constitution doesn't forbid the kind of eminent domain exercise as happened there, it also doesn't prevent state and local governments from enacting laws that restrict takings and prevent commercial involvement, or simply impose an arbitrary time limit in the hypothetical examples above. But those are not constitutional issues; moreover, the most zealous critics of Kelo generally want to minimize federal government's powers and favor maximal freedom of state and local governments to run their affairs as they see fit. Yet in this case, they wanted the federal government - federal courts, no less - to invent a new concept in the Constitution and use it to tell every state and local government, for every property transaction they make, whether they are allowed to do so or not.