As you say your garden isn't very big, could you go for a low maintenance style garden? Hitting the ground elder with weedkiller as suggested would give you a good start before doing some major hard landscaping.
No, don't dig it up because the ground elder roots are fleshy & very brittle. They break up & each little piece will grow to become a new plant. Covering the ground with a weed proof membrane will help after the first few proper die-backs of the elder. Excluding the light will effectively finish the job of killing the elder once the top growth is killed off.
Then employ your gardener to lay gravel & or paving to create an area to walk & sit. Containers & or raised beds with imported soil would also give you a weed free 'instant' start but will be expensive. You will get better value for money investing in the bones of your garden rather than chipping away at it. Lawns, while lovely to look at, require a lot of care & time. You would have to ensure that practically all the elder was killed before trying to establish a lawn from turf or seed.
If you check out some of the best 'instant garden make overs', the most successful start with the least promising sites but require really good hard landscaping. Keep asking - we'll keep answering!