I have not done any desktop work with AppGameKit recently, but a couple years ago I created a NSIS installer script that added the minimum required DirectX audio component to resolve this (it adds about 2 MB to the installer). You will need to update the path to where your DirectX setup files are located to bundle into the installer. You can't bundle anything directly into the AppGameKit EXE but need to create a full installer for dependencies.
; Install DirectX Files for AGK Compatibility
Section "-InstallMinimalDirectX"
SetOutPath "$TEMP"
File "D:\DirectX\Jun2010_XAudio_x86.cab"
File "D:\DirectX\DSETUP.dll"
File "D:\DirectX\dsetup32.dll"
File "D:\DirectX\DXSETUP.exe"
File "D:\DirectX\dxupdate.cab"
ExecWait "$TEMP\DXSETUP.exe /silent" $0
Delete "$TEMP\Jun2010_XAudio_x86.cab"
Delete "$TEMP\DSETUP.dll"
Delete "$TEMP\dsetup32.dll"
Delete "$TEMP\DXSETUP.exe"
Delete "$TEMP\dxupdate.cab"
${if} $0 != "0"
MessageBox MB_ICONEXCLAMATION "DirectX may not be properly installed on your system. If you experience problems running this application, please download and install the DirectX runtime files from http://www.microsoft.com/en-us/download/details.aspx?id=8109"
${endif}
SectionEnd