First off this sounds totally normal both my DDs were like this in KS1 and Y3.
Second - learning to listen is a skill - and like anything else it needs practise.
Nature watching (which requires you to be patient, quiet and pay attention is a great way of practising these skills). Spring Watch should be coming to the BBC fairly soon (it's on most years) - springwatch activities here: www.bbc.co.uk/thingstodo/project/springwatch
With DD2 who rarely sat still and is constantly distracted - teaching her to sit quietly and wait for the butterflies and take photos of them was really useful.
In the autumn we looked for mushrooms and collected fruits like blackberries and bilberries. (again requiring focus).
If you have a bird feeder - encourage your child to record what birds come to the feeder. www.rspb.org.uk/birdwatch/
Try also watching things like documentaries (often on a bit late but you can watch them from iplayer or recording) - and encourage your child to just listen.
Go to story telling events at local library, etc... - again encouraging listening.
Encourage chores: So simple instructions at first. DC could you go fill a watering can and water my flower pots.
DC can you take these clothes upstairs and put the shirts on my bed and then put your school clothes away in your drawer. (DD2 used to leave everything on my bed).
It may also help to take stock of how you behave when doing reading with your child. Is it quiet and calm, focused on the reading - or is the tv on, radio blaring, lots of noise/ interruptions. If your DC is used to that - they won't really be used to being calm themselves.
Now some kids just have tons of energy and adding fuel to that (in the form of sugary foods) can make them even more energetic and less able to settle down and get on with the task. So one trick my brother (who teachers in primary in the US) uses is to encourage children who are a bit overactive/ inattentive to make healthy eating choices and eat more fish. (He genuinely sees an improvement when kids get off the fizzy drinks and eat less sweets).
HTH