Welcome to the forums!
You have certainly come to the right place as you will find all you need to know on budgie care here.
The information on the several articles and sticky threads at the top of each section of the forums will be extremely useful and of benefit to you and London.
Before allowing London some out of cage time, it's important that you bird proof the room so that it is safe for her.
Given her older age and the fact that she hasn't been let out for a considerable amount of time, it would also be good at first to confine the area where she will be having out of cage time in and if possible to place some netting in strategic places in order to prevent her from crashing into a wall or piece of furniture.
It would also be good to reconnect with your budgie first and work on re-establishing the trusting bond to later teach her to step up to your hand/finger.
This will make it easier to take her out and place her back in the cage after the out of cage activities.
There are ways to encourage your budgie to venture out of the cage, you can do so by placing a play area near the cage, having the cage's door open and attach a familiar perch on the cage's bars (outside part of the cage and near the door).
Placing a a couple of your budgie's favourite toys on the play area and a tasty treat can also do the trick.
For al the detailed information on this subject, be sure to check the stickies at the top of the Taming and Bonding section's main page.