Yep , you could pay hundreds on an exterminator ( as Martin says ) but the chemicals they use are as indiscriminate as the off the shelf ant killers we can buy and probably more so , I would not worry about other species as with the amount of ants you are describing the other life forms would have been long ago eaten , ants are like bees in their nest structures , workers , soldiers , drones , queens etc .
On that the best time to start hunting the nests down is as the sun is going down , the last few workers are returning home where the army of solders is defending the nest and they are the ones that bite ( it sounds like you have a nest by the cloths line ? if you are getting bitten there as the soldiers don't stray far from the nest ) , its quite easy to find the nests but it just takes time , I actually like observing them and following them so I can nuke the nest , as the exterminators will do for a cost but its your money .
At home in NZ we have a real problem with the German wasps , huge dangerous nests under ground humans and dogs have fallen into them and died " Painfully !!! " ( they are just like ants ) and what we used to do is catch a wasp just on sunset and tie a 200mm length of cotton to it and let it go , and they always go straight to the nest , easy to follow and nuke it just on dark when the hive is full , take your time and find all nests and their multiple exits , I use posts in the ground as markers , just like the experts do , its a big job to do properly as one nuke wont do it , yes it will slow them down but they will be back, just like Arnie
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One more thing , my front lawn was like yours only 12 months ago ( its terrible , or what ? ) and its almost ant free now using the method I described , yes there are some but they are under control now and there is also many other species around now the ants have been sorted that were not there before , its all about balance
Enjoy .
Brian.