For typical viruses on a Windows computer, running a good system inspection and virus removal product (such as your running of Kaspersky) will be sufficient. A complete re-install of a system due to virus infection is usually a last resort step due to There is nothing that can save this computer reasoning.
However, there are still some very small subset of viruses that can linger on the Boot record of a hard drive which a typical re-install of Windows will not get rid of. You would need to format and recreate the boot partition, or use a rootkit virus scanner.
Another infection vector includes websites or already compromised files that you might restore to the new 'clean' system. Be sure to inspect (virus scan) and clean any files you move over from the infected system before opening or using them
Programs or software I have found useful in detecting and removing viruses from 'hosed' Windows systems:
Question: Will the re - installation disable any lingering viruses or should I format the whole hard disk to get rid of the viruses completely?
Short Answer: Typically, yes. However certain viruses may linger after a re-install, or may be reintroduced to the system after restoring from backups.