You don't have to use agencies - though there is usually no charge to you to do so, until you decide to hire someone introduced by the agency.
As NannyL already said, www.nannyjob.co.uk is one of the most popular sites that nannies look on - at least, its where I have looked for work in the past. However, my current job was listed on www.nannyselect.co.uk so that may also be worth a look. Also, contact your local newspaper to discuss classified advertising rates - as you want someone live-out and thus someone local.
Flexibility on days I feel may become an issue... most nannies in my view will want a guaranteed income. You may be lucky and find someone prepared to do what you want, but if finances allow you may find you get more applicants if you can say the job is Tues-Thursday every week. Unsocial hours should not be that major an issue, though of course some people find getting up early/working late hard. Unsocial hours however may increase the salary a nanny expects.
Typically budget for £100 per day based on a 10 hour day, so £10 per hour (to include employers NI). This is not the take home pay the nanny gets (so don't put it on adverts). Roughly - based on 3 days, £300 cost to you, is about £275 gross for the nanny. However many factors could affect pay rate you need to offer - such as unsocial hours, duties you require, what applicants expect etc. IMPORTANT: always agree a GROSS wage (before employees tax/ni deductions), never Net (take home).
Local authorities are being advised by Ofsted about nannies who register under the Ofsted Childcare Register - so you may find that your local authority does know of some nannies. However, it is not under their remit until Sept 2008, thus you won't find this information on childcarelink.