I'm quite sure that you should not dilute the HClO4. I have 60% HClO4, and even when this is mixed with potassium iodide, no iodine is formed, so forget about the periodate with dilute HClO4. HClO4 only works as a strong oxidizer, when its concentration is very high, and even 70% HClO4 only is a very slow oxidizer, even towards easily oxidized compounds like KI.
KMnO4 indeed is not a strong enough oxidizer to make KIO4. It is even the other way around. Periodate is capable of oxidizing Mn(2+) ions to permanganate quantitatively at reasonable speed and only slight heating (50 C or something like that). This is really amazing, I know no other aqueous oxidizer, which can oxidize Mn(2+) to permanganate so easily as periodate can do.
Also keep in mind that KIO4 is only VERY slightly soluble. I have NaIO4, which is soluble much better. KIO4 hardly can be used in aqueous solutions, unless a slow release oxidizer is needed of good strength.
One way to make periodate probably is electrolysis of an iodate solution. In a similar way, perchlorate can be made from chlorate. I have no practical experience, however, with making periodate. I just purchased it for a few bucks for 100 gram (from an old GDR lab). Yes, even home-chemists can find this stuff