You can cut Trachelospermum jasminoides hard back. It will not mind in the least.

Thank you. Does it matter in when this occurs and do you know how quickly it might recover?

In my climate I'd do it in spring but in yours you could probably do it any time. It should regrow several feet in the next season. Thereafter you can keep it more under control so it doesn't get out of hand again by regular trimming out of stuff you don't want and guiding of stems to where you want them.

One way to lessen the visual impact would be to cut down half this year and half next, although this would involve a good deal of disentangling by the sound of it.