Try Weedol for lawns. This reduced it considerably on our lawn but it is persistent and will probably need repeating next year. I see there is now a Weedol Extra Tough but haven't tried it.
I don't think there is anything else you can do but maybe somebody else will have a suggestion.

I think it's called self heal and unless you keep it under control it will almost take over and spring up in your borders too. A good lawn weed killer and fertiliser should help though you may have to treat it two or three times.

Definitely persistent. Self heal, gets into the borders as well. Sadly you tend to spread it when you mow the lawn too. Weedol does reduce the incidence, but you have to keep on attacking it, for probably years.

Basic tips for using selective weed killers:
Mow and water the lawn 2-3 days before treating it, to encourage active growth of both lawn and weeds.
If it is time to feed, feed at the same time.
By contrast, "weed & feeds" are usually less effective on weeds, and more harmful to the lawn, than liquid weed killers.
Spray so that the weed leaves get a thorough coat of killer, but not much drips onto the soil, which can damage surrounding trees and shrubs. If the formula doesn't already have it, a surfactant helps with that.
Unless your neighborhood habitually gets heavy dew, spraying in the evening is most effective.
Don't water--and hope for no rain!--for 24 hours after spraying.

Its usually too windy here for spraying. I do ours with a watering can and believe me it takes a long long time....