http://www.r4isdhc.com/rtswhite/
You want the flash kit firmware update (this will fix compat issues with the latest 3DS firmware)
You do still need a DS or DSL to update it though if you have updated the 3DS before updating the flash kit, either find a friend with a DS or DSL.
Why you went for that flash kit is beyond me but each to there own, if you need to buy a new one though I recommend you ask them if they will update it before shipping it out to you.
You could ask someone online to update it for you, but that is a big risk to take. (If you do though ask for someone that lives in the same country as you)
The kernal is just the software on the Micro SD card, make sure that is on the micro SD though and test again, you even need that just to run the firmware update.
The kernal is not for making it run on 3DS firmware updates, only the flash kits firmware update can do that
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