As it turned out after checking comments adding wl.skydrive
to Scopes property solved the issue:
<live:SignInButton ClientId="my ID" x:Name="btnSignin" Scopes="wl.signin wl.basic wl.skydrive"
Branding="Skydrive" SessionChanged="btnSignin_SessionChanged" Margin="10,0,-10,194"
Height="104" VerticalAlignment="Bottom"/>
More about available scopes you can read here at MSDN.
Also your App should me marked as for mobile devices at your account.