Unfortunately both ivy and bindweed are never killed off in the first treatment as they have good root stores and will regrow again (usually more vigorously than before - a natural response to replace lost foliage quickly so they can photosynthesise and build up food stores in the roots) You can keep cutting the plants back where they grow at soil level before they get a chance to grow into plants again and eventually they will weaken and die, but you really need to keep on top of it and sometimes over 2 years. The moment they get a hold again you are back to square one I'm afraid.
I don't mind using weedkillers etc but they must not be harmful to other plants or fish.
The bad news is that ALL weedkillers will harm other plants and fish as they are designed to be herbicides (which in pond water will quickly kill off aquarium plants) but if careful, you can treat just the plant you want to get rid of. If you have to use a spray, cover the other plants/lawn/pond you don't want to be accidentally contaminated with a sheet of plastic (like the ones you can get for protecting carpets and furniture when home decorating or use several sheets of newspaper) where stray spray droplets will fall, and make sure you only spray on a dry (essential as you don't want the chemical diluted or migrating outside the spray zone), calm (windless) day and aim close to the leaves of the ivy/bindweed. The weedkiller should be dry within an hour and be safe to remove the plastic sheet afterwards - take precaution and fold the sheet so that any possible contaminated side is shielded from other plants although to be fair, most weedkillers that land away from plants usually become ineffective by then.
Alternatively, you can buy weedkiller in a gel stick like this
link so you "paint" the leaves (make sure other plant leaves don't accidentally brush against the painted ones)
This advice from Roundup:
Ivy is a particularly problematic weed as it can be resistant to many herbicides, and if left it can end up taking over your garden. In order to deal with ivy effectively, it is necessary to use a weed killer that targets the entire plant in order to get kill it completely. The best time to use a weed killer for ivy is during the autumn as weeds will be starting to turn dormant and will be storing nutrients. By using a spray at this time of year, the weed will store the weed killer which will give you the best results and enable you to control the ivy better.
Bindweed - they have long brittle roots which readily grow into into plants when damaged, and easily grow where you don't want it to. A neat trick is to push a cane/stick into the ground near where you see it growing out of the ground and train the vine tip to climb up it. You can then use weedkiller on the plants growing up the canes.