Adding a few thoughts here, since I've used PCS since DOS version two! First, it used to be that a cheap laptop would suffice, A slow processor wasn't a problem then.
But ever since most of our apps are web-based, I've found that screen readers take a big processor hit when they have to continually refresh the offscreen model. So when I've had to use a laptop with a slower processor for the past ten years or so, I've found it very painful. The response time of the screen reader becomes very sluggish on any website that has any interactivity. Also Office is dreadfully slow.
So even though you see all these great deals, laptops priced at $200 (that's American dollars) avoid them like the plague. I think it's better to save up for a while and get something with a processor that's fast enough to enable you to use all modern applications without latency.
I think I'm less worried about the keyboard than many folks. I decided long ago to get cheap desktop keyboards and connect them to my laptop at home, and a Bluetooth keyboard I really like for traveling. So I just don't use the laptop's keyboard.
Or at least I didn't until my employer gave me a Dell Latitude 5420. I just love its keyboard. It's a business model, and clearly, Dell realized people would be doing extended typing on it. And it's just a 13-inch screen laptop -- very light. And fast. On the open market it is $1300, but likely my employer buying in bulk, paid a lot less. 
I echo Brian's comments about getting a refurbished business model if your budget cannot take a new one. And there's a lot less crapware on them as well.
Laptops I bought from consumer stores were full of bloatware software I didn't want and had to uninstall. The business Dell though had only a few small apps I needed to remove. 
I also echo the thing with RAM. If you don't have enough, each time Windows needs to use virtual memory -- ram content cached to disk, it gets super slow. And if you have multiple apps open, it's a huge pain. If you tend not to multitask very often you can get away with less ram, but right now I have several browser windows open, Cisco jabber, Outlook, a word document and a spreadsheet.  On a desktop you can get someone to easily and fairly cheaply add ram but it's not so easy or inexpensive on a laptop, which is why the manufacturers know they can charge more for increasing the ram on a laptop. 
My husband is an electrical engineer, so I bought a laptop once that had minimal ram, but I knew it was user serviceable and he could purchase and add more ram for me at a much lower cost. If you don't have a pet engineer handy, you have to pay more, sadly.
Lastly, remember too that screen readers themselves consume memory especially if they are dealing with very interactive apps.  
I'd also say arm yourself with a good set of wireless headphones so if you type for extended periods you can easily get up and move around. It's not good for your body to sit for hours.