If you flash on command line, then the last update is R21.7:
[url]https://developer.nvidia.com/linux-tegra-r217[/url]
(you might have to login and then go to the link again since URL redirect does not always work)
FYI, this is the final release for 32-bit systems. After that 64-bit is required.
You can compile natively on the TK1. You will also find the “Documentation” download (within the driver package page…this is the “L4T” flash package) has the 4.x version of gcc cross-compiler which is recommended (the “Documentation” is an entire html directory tree…the “baggage” subdirectory is where some additional files are found).
That is the correct file, but it looks like R21.5 did not yet include actual cross-compile binaries. Instead it has an appendix which lists the configuration for building this yourself (which is not simple). However, you are still in luck because the TX1 64-bit system was initially supplied with a 64-bit kernel and 32-bit user space…and so the 32-bit compiler from the TX1 downloads works for what you are doing. Go here:
[url]Jetson Download Center | NVIDIA Developer
Among the entries which show up look for the 32-bit R24.2.1 listing. This is the same as what the TK1 uses. Listed as “GCC Tool Chain for 32-bit BSP”.