Stephanotis floribunda.Syn.Jasmine Stephonitis It'll respond well with outside air nothing to hot for temps or to direct from sun for to long if to go outside.
The water you give/gave it isn't being taken up by the root.
Water could be perched or has compacted in the soil and rotted some of the roots.

Mite problem: sounds like you have a very good plan for them guys... As most insect soaps are mixed or already to use premixed with water, wouldn't watering have a slightly twisted trick if your not suppose to water ?

As your watering habits would need to change consider why not change the pot and the soil as well ?

Dizzy,
Do you have a spot outside that receives afternoon shade?
The jasmine you have will appreciate a little shade in the hot afternoon.
I would put it outside.
Do you want to keep it in a pot, or plant it in the ground and dig it up in October and bring it in?
If you want to keep in a pot, you are going to have to bump it up in size a bit, not too big, just a size or two bigger.
Change the soil too. Check out the roots when you take it out of the exsisting pot, see if the are circling a lot around the outside of the soil when you take it put of the pot.
If they are, get a sharp knife and gently cut the roots a little all the way around the bottom of the soil before you put it in your new pot with your new soil.
By cutting the roots, this will encourage the roots to grow into the new soil in its new pot.
You water your plant when it needs it.
Stick your finger down into the soil as far as you can, see if it is moist. If now, water it. The hotter it is, the more often your pot will need soil.
Good luck with your new vine, it will be fine.