You will never be able to eliminate "weeds" and keep them gone because they tend to seed very prolifically and will fill in anywhere there is space for them. No matter what anyone tells you about some product none will eliminate the need to work in the garden to eliminate these unwanted plants. The most environmentally friendly, ecologically sound, method of "weed" control is to mechanically remove them, ie. dig them out. Beyond that the products sold to "control weeds" are being found to do serious harm to our world.

The info in this discussion should be helpful to get a handle on this sad little garden. Please ask if you still have questions or get confused trying to relate that to your spot. The first thing I'd do for this spot is decide where the grass ends and the bed begins and put some kind of border, like bricks or a landscape timber to define the separation. The flowers are pansies, short-lived and usually killed by the heat of summer, not worth saving.

And mulch after you weed - it will make a huge difference in how often you need to weed (says someone who needs to remulch everything).

Use a fork and dig up all the grass and pansies, shake out the soil and start over. That's all you can do right now.