A lot of fun, landscape and museum, coffee and more
You and the kids will love this please, The Pearl, you will see it on the hill from the balcony https://www.perlan.is/en-gb
Birds & play ground
The Reykjavik Pond is few min walk where there are some items and ground for kids and on other part of the pond they can feed the birds https://goo.gl/maps/X7AFrLtpxr1rvADk7 you will see the City Hall there.
Playground, coffee, restaurant
Nice area also here for kids, some play items and nice outdoor fields area, coffee and restaurants as well in 10 min walk https://goo.gl/maps/WqvZHhRqMYANm97E7
Restaurant for brunch
There is a restaurant call Laundromat in Austurstræti, I think there is still a kids area in the basement, brunch all days. 10 min walk further downtown. Walk Skólavörðustígur down till the end and to left and walk until you see it. https://goo.gl/maps/pKNuKeHsMWbS4CwTA
This is my favorite pizza, two min away in front of the playground on Freyjugata, both fine to sit it and also takeaway https://www.reykjavikpizzeria.is/ location https://g.page/reykjavik-pizzeria?share
There is small coffee place very close to you which is called the Cat coffee but there are few cats there are very friendly when having young kids, only few min. https://goo.gl/maps/mU9pVaE95ewAMr4B7
You all might also like the whale watch museum https://www.whalesoficeland.is/
There is swimming pool on the hill, the other side of the church, 3 min away, very nice but also others on the Reykjavik area which might be more interested to the girls https://sundlaugar.is/en/landshlutar/capital-area/
Zoo & more, restaurant for lunch, swimming pool
This here is a must for young kids, to visit the zoo.
The place is close to where they take the covid test if you need to do that so I recommend to combine when you do that, you can take a taxi to the covid test and walk to the Zoo https://goo.gl/maps/8ZCcnteby6SxwMXj7 14 min walk from where they do the covid test
In Laugardalur you will find one of the most popular outdoor swimming pool in Reykjavik, you will see the national football ground.
We have a special area there called Grasagardurinn which you can reed all about there https://grasagardur.is/en/
We have there on the Grasagardurinn area the old washing pools but under the ground of this area there are spring holes and before the hot water was distributed to houses, women's used to walk from the area where my penthouse is and even further all the way to this place with the laundry, wash it and go back in all kinds of weather, you will see on the area old relics about it, can read about it here https://vatnsidnadur.net/2019/05/24/thvottalaugar/
On this area we also have the family and zoo park, where you can see most of all the Icelandic animals https://mu.is/.
There is a nice restaurant within all the trees and plants where I recommend you have the lunch this day but in case I recommend you make a reservation http://www.floran.is/floranenglish#cafebistro-english