Tried sprinkling a wildflower mix, nothing grew. Maybe the seeds were duff. Most wildflower mixes are heavy on such things as cornflower, poppy. These are annuals, and because they only last a year, they don't build up enough body to compete against grass. So to ensue they get the bare soil they need, they are often triggered into germination by light - an indication that the land has been ploughed and they are lying on the surface of bare soil.
So the first step is to make sure your seeds are for the mainly perennial plants that grow in grassland.
Even then, they'll struggle to get away if scattered on a dense sward. So the two approaches are:
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Strip the grass and get back to bare soil, then sow seeds
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Grow the seeds in pots, or buy plug plants, and when large enough plant them into the lawn, ideally clearing a space around them to give them chance to get established before they have to compete with the grass.
There are broadly two strategies for wild flowers a) get big and tough so you can out-compete anything b) develop a tolerance to inadequate growing conditions, eg shade, dry soll, boggy soil, or lack of nutrient.
Most grasses, along with nettles, docks, thistles, use strategy a). With plenty of nutrient the grasses will win. Most "desirable" wild flowers use strategy b) So to establish your wildflower meadow, you need to tilt the balance in favour of the flowers by reducing the nutrient level - in other words:
Ideally, start by stripping off the turf and top few inches of soil, where the nutrient is concentrated.
Unless you've planted spring bulbs, keep the grass short till the beginning of May, collecting the grass clippings and removing them.
Then leave un-mown till mid July when the grasses have peaked
Start mowing again after mid July, taking away the grass clippings.
Sow yellow rattle, which is semi-parasitic on grass. It needs sunshine, and it needs sowing fresh, and making sure it's in contact with the ground and not stuck in the thatch.
In hay meadows managed for plant diversity, road verges etc, cutting and removing grass while it's still in peak condition is more important than allowing the flowers to seed. The flowers are perennial and will still be there, whereas once the grass is going over it's busy returning nutrient to its root and to the soil, allowing it to compete better the following year .You have some flexibility in a garden, but leaving the mowing till October year after year will reduce the flowers and allow coarse grasses to flourish.