I live in Ottawa Canada. My flower bed is full of a specific weed (see picture). I've tried pulling it out, it comes back stronger. I've sprayed it with soapy water, its now clean. I've sprayed it with diluted vodka, its now tipsy. Is there any way to get rid of it, bar a flamethrower?
Outside United States
(New Brunswick, Canada, here... nice to help another Canadian...)
I can't be 100% positive in the ID, but from what I can see and from your description, my thought would be the dreaded goutweed (Aegopodium podagraria), which is an extremely invasive and tenacious plant that spreads by underground stems (which is why it got stronger when you tried to dig it up... you effectively created lots more plants. Neither soap not alcohol have any herbicidal properties so would not be effective. Only strong methods over a period of several weeks will get rid of it, and whatever you do the application has to be 100%. If you leave behind one stem or one plant, it'll all come back again. It is also unfortunately the case that anything you have in the bed you *want* will be lost in the process. Do not try to pot up other plants before treating the area because it is almost certain you will also dig up goutweed roots and simply transplant the problem. To solve the goutweed, you'll have to sacrifice the flowers, for now.
Not sure about restrictions on herbicide applications in your location, but Round-up or another glysophate product will take care of it and is the safest, non-persistent herbicide. You only need a fine misting on the leaves. Since it only impacts plants that are in leaf at the time of application, you may have to re-apply a couple of times as succeeding growth appears. But it will take care of it. Glysophate products applied according to directions in these sorts of extremely limited and specific situations have no negative environmental impact, but many municipalities have banned their use.
Do not waste your time or money on so-called "natural" treatments (eg. vinegar, epsom salts, and so ... or even flaming...) these at best will only knock back the top growth, leaving the roots happily in place to continue growing.
If you can't use glysophate where you are, the next easiest approach is called occultation... fancy word for covering the entire bed with heavy, light-blocking material (cardboard, landscape cloth, tarps, etc.) which blocks light and you have to leave the material in place for at least 6-8 weeks. With the warm weather coming this method can be very effective. But you need to cover the entire area and leave it undisturbed until the plants are completely dead. At that point you can remove the material, work up the bed and replant.
Other than that, the only approach is to remove the soil entirely to a depth of 6-8 inches and replace it with new growing medium. This is of course labour-intensive and expensive, and it likely simply spreads the weed somewhere else wherever you end up dumping the soil. I would not recommend this approach.