Dear Arindam,
What I understand is, your system has two Ubuntu 12.04 installations side by side. Are you using any one of those two to compile the 2.6 kernel? If so, then executing "sudo make install" will copy the kernel vmlinux, initrd and System map files to /boot directory and trigger an "update-grub" to rebuild grub menu. Your new kernel should appear afterwards. May be you are not being able to see the kernel listing because of having a ZERO timeout in GRUB menu which is causing it to boot to the default one directly. Make sure it is not the case.