"Walking would be great. Green space is ideal, and we’d want at least a balcony to sit and have a coffee."
30 mins walking with green space and a balcony is a bit of an ask. 30 mins commute isn't so much the problem as 30 mins walking. The Barbican would probably be the place you're looking for (see below)
If green space is what you're after then start off by looking for flats near green space rather than perhaps for particular areas (although there are areas with a lot more green space).
For example this flat which is 2 bed, furnished, 834 sq ft with a balcony and looks out over St John's Gardens in Westminster for £2,750. Also has 24 hour concierge.
It's a 10 min walk to the tube station then a 20 min journey on a direct line on the tube to Moorgate.
https://www.rightmove.co.uk/properties/147035486#/?channel=RES_LET
Also, don't forget that in the UK it is the tenant that is responsible for paying property tax. The property tax (it's called Council Tax here) for that property is £117 per month.
A previous poster mentioned "or the city itself". The Barbican can be a very nice place to rent. There is one available at the moment for £2,750, 2 bed unfurnished 840 sq ft. Council tax is £124 per month. You've got a big green space on one side and tennis courts on the other.
The Barbican is literally only 15 mins walk to Moorgate.
https://www.rightmove.co.uk/properties/147129065#/?channel=RES_LET
Heading out east, you can sometimes find flats overlooking Victoria Park
Here's one, 2 bed furnished, 795 sq ft £2,600. Council tax is £124. It's about a 20 min walk and then another 20 mins on either bus or tube.
https://www.rightmove.co.uk/properties/146948279#/?channel=RES_LET