In making solar power, one important concept to understand is what's called a "peak sunhour" (sh), which is basically the amount of "FULL" sun you get per day. You multiply your solar panel size by the sunhours to get the total amount of power you can make in a day. For London, at 51 degrees North, I think you'll get approximately 1.5-2.0 sh per day in December, and 4.0 sh in June, in full sun. On cloudy/rainy days however, the sh is likely to drop down to ~0.5sh.
These numbers become centrally important with a whole-home solar system, where you're counting on the system to make ALL your power. I'm working with neighbors down the hill from me right now because they can't understand the idea that there are less sunhours in December, and they have to deal with completed dead batteries right now.
A second factor to consider is that the rating of your solar panel is measured in a controlled temperature chamber with artificial sunlight scaled at exactly 1000W/square meter. Your real-world solar output is almost always somewhat lower, because of dust in the air, less than optimal angle to the sun, and how hot the panel is. So, for real-world output, I usually recommend 85% of what name-plate production is. That means that your 10W panel is likely at best to make only ~8.5-9.0W. On a cloudy day, maybe just 1W.
When you want to power anything with an electric motor, you also have to contend with something called "starting surge", or "inrush current". For something like a free-spinning window fan, the starting surge might only be 1.1X the running power, but for motors starting under load, the starting surge might be 3-4X the running power, for maybe 500-1000 miliseconds. I'd consider a blender an appliance that would have a significant starting surge. With too small an inverter, and too small a battery, that starting surge might drive the inverter into low-voltage shutdown.
What that means in the real-world is that people are always overestimating how much power they can make, and underestimate how much power they are consuming. With what you are describing, I'd recommend you'll more likely be served with a 200W panel, charging a 100Ah battery. Because an inverter is also a power drain on your system, look into powering all your needs with DC-only appliances that can bypass needing to make AC power.