Does anyone know a way to get rid of mildew, and when to do it?
Prevention is better than cure - mildew attacks when the honeysuckle is under stress, usually because of dryness at the roots. It needs lots of water regularly - give the roots a good soaking in hot weather - honeysuckle likes it's roots to be in the damp and the shade.
Once the honeysuckle has got a bad attack of mildew there is no treatment that I know of that will cure the affected leaves. I let those leaves die off - if you improve the conditions at the roots it will soon grow new shoots and leaves.
I recently bought a reduced price honeysuckle from the 'nearly dead' section in the garden centre - the watering had been neglected and although it looked healthy when I bought it within a few days it had succumbed to powdery mildew and lost most of its leaves - now, with plenty of water at it's roots (at least a bucket full every day in this dry weather, taking care that it soaks in and doesn't run off the surface) it's growing healthy new leaves.