surrey is semi near me, asim on kent border but you are a tad too far for me (sorry)
as you have time you can try putting an ad on childcare.co.uk (rem to pay and be a gold member so you can send messages) or nannyjob or gumtree
you could ring local agencies and enquire if they have night nannies, cost varies but generally £25/30 a night on top of what you pay the night nanny (which can vary depending on their exp but around £12/15 gross an hour) or as your 2nd child you could have a nn with less exp and pay less
generally one night a week isnt enough if you want the nn to implement a routine and give you a break - i recommend 2/3nights a week to begin with and then drop to 1/2 a week
work out your budget and then a nn will help you get the most from it
will you be bf/bottle, or ebm
if bf do you want nn to come to you with baby, leave/talk to you, then after settle bubs and you go back to sleep
if you are happy with a bottle, i would try to do the night feed as one, either formula or ebm, so that you get a proper nights sleep and dad/gp/nn can feed and help you out