Basic question - LM833 single power supply
Well I think you have most of the answers now so try this: Use a split rail supply i.e. +/- 5V or greater (not less) with the centre point connected to the arduino ground. This means the op-amp receives signals that are centred to its power rails. Disconnect the sound card input and check that it isn't warming up. If it is throw it in the trash and fit another IC if you are confident it is wired correctly.

When you are happy with this stage connect the ouput to the sound card via a 1kohm resistor and hopefully, it'll be fine.

Don't try running it with a total supply rail less than 10V because there is no guarantee it will work. I might also be tempted to put a 1uF capacitor in series with R1 (on the arduino side) just to prevent any DC getting into the input. If you do do that you'll need a 10k resistor down to ground straight after the 1uF.