The thing is, you need to have a pretty brutal look at the costs and commitments of adding to your family by getting a dog. Because if you start looking at actual dogs first, your heart will melt and you might not make the best decisions.
It does sounds like you're in a good enough place to get a dog, and your DD is old enough to do a share (and she can help guilt her father into keeping his promises, but if he's usually reliable then you should be OK)
Is the number of days in your new workplace fixed and long term? Because if not, then make sure you can afford enough daycare (dogs at work is way less common than it used to be)
Think carefully about not just size, but also how energetic (breed traits a useful guide, but never guaranteed for the individual dog), how much grooming, how much drool?
If you really want a puppy, you will need to have a lot of time - the foundations you lay will make an enormous difference to the dog you have. So doggy daycare rather than a visiting walker might be needed until the pup is older.
Also, if considering a puppy, have a look at the vulnerable and at-watch native breeds
www.thekennelclub.org.uk/getting-a-dog/are-you-ready/vulnerable-native-breeds/
These tend to come from breed enthusiasts, not larger commercial breeders, so you may need to wait. For small friendly family dogs on those lists, try the Norfolk or the Norwich terrier.