Mildew actually damages the fibres, so it's not certain you can ever totally remove the stain. The best removal method is still bleach, but that can damage the fabric too, and result in the strength of the sling being reduced, so start off with dilute bleach and increase strength until you get a result. Wash out with clean water many many times to remove any remaining bleach in the material - you don't want it to continue being active after stain removed.
Don't forget to test if the bleach removes the colour - you could end up making things look worse. If you just want to stop the mildew getting worse, give it a good scrub, then soak in full strength Dettol. Don't wash the Dettol off, it will provide some protection against future growth