I bought a baby monitor which came with a 2 pin power adapter
In the UK? It should not have been sold with a 2-pin power supply unit unless that was fitted with a permanent, approved adaptor. Even supplying a separate adaptor is not sufficient. The supplier is technically breaking the law. If you bought it from overseas, caveat emptor.
Personally, I would put it to one side and buy a UK approved 5V 1A power supply unit (I think it says 5V on the label) with the correct output connector to suit the baby monitor itself. Units are available with selectable voltage and polarity and interchangeable tips, that will suit almost any device. Buy from a reputable, UK-based supplier, the cost is modest but the safety is much less likely to be compromised.
However.
If your existing power supply unit has two round sleeved pins like a Europlug, it is safe for 230V by UK / EU standards, although not ideal to connect it using a plug adaptor. If on the other hand it has two flat unsleeved metal pins (US configuration) then it is not recognised as safe for use at 230V. Although the label says it will function on this voltage, the protection of the pins is not adequate and could create a risk of severe shock, so I would not recommend using it full stop.
Adaptors sold in the UK fall into three main classes. There are permanent, approved convertors that fit onto (e.g.) a Europlug and turn it into a BS1363 plug. These are safe, compliant and there is no reason not to use them. Then there are travel adaptors made to recognised standards such as BS5733 and BS8546. If these are fitted with a compliant fuse they should be safe for low power devices like your baby monitor, bearing in mind the paragraph above. They are only really intended for temprorary use. Finally there are vast swathes of non-compliant travel adaptors that violate so many regulations it's difficult to know where to start, and besides not being technically compliant, they are in real life genuinely unsafe to use due to risk of shock or fire.
help me determine what amp size my fuse should be
The load current of the adaptor is shown as 200mA (0.2A) but this will be the maximum at 100V (the lowest rated voltage). At 230V the load is likely to be less than 200*100/230 = 0.087A. Even if the switch-on surge is 15 times the running current, it will not blow the fastest 1A fuse. Therefore a 1A fuse is likely to be sufficient. However, if it is approved for the EU market and fitted with EU style plug pins, it is designed to be plugged into a socket protected at 16A, and will have an internal fuse to protect itself. Thus, any rating of fuse
within the rating of the adaptor would also be suitable for this device. Again, assuming it is correctly approved for the EU.
More important is the kind of fuse. I have seen glass fuses and all sorts of inappropriate types fitted to adaptors, which will not provide adequate protection at all since they are often not rated to stand the high fault currents of a UK socket-outlet circuit. The fuse must be made to BS646 (the smaller size often found in adaptors) or BS1362 (normal plug fuse). Nothing else will do.
Don't use 13A. The cable won't be rated for that
I don't think there is a cable on the 230V side (the OP implies the pins are on the unit, and it's only 3.5W output.) If there is, then if made for the EU it will be 0.75mm² minimum and OK with a 13A fuse. If not made for the EU, then it may not even be approved for 230V so should not be used. The output cable only needs to handle 0.7A at 5V.
I've gone into this in depth a) because I'm taking a break from grovelling under my hot water cylinder and b) to illustrate that the initial premise - that choosing the right fuse rating would make everything OK - is not necessarily true!
