Budget is not a huge deal but around 500 euro per month would be perfect
That is an incredibly difficult question. Apartments are difficult to find unless you sign up for 6-12 months. Airbnb in Bali rarely stray away from the Tourist locations. There are a handful of places away from Tourist hotspots that are either poorly rated or are well above your budget. I'm making the assumption that you haven't been to Bali for the purpose of this answer.
(Side note, it is near impossible to avoid Tourists altogether on the island unless you want to live in a location that it'll take over an hour to get to any decent amenities - but then again it is rare for anyone to have "nice" accommodation far away from western comforts. Tourists are everywhere on the island in varying levels. Hopefully you mean away from tourist hotspots).
What is your definition of 'good WiFi'?
Bali's Internet has to go through Java, which in turn goes through various spots in Asia depending on the Internet provider before connecting to the rest of the world. I see you're from Leeds which averages 30Mbps. Cities on the island are lucky to get 3Mbps. Beach locations away from tourist hotspots may only get half as much. I've stayed at places out of the city centres that run off 3G hotspots that constantly drop out. OK for eMail and that's about it.
But I liked the challenge of this question so I gave it a go - this is what I found:
La Kubu actually looks quite nice. Fairly standard style Bali villa with the usual trimmings and close to the beach. I've actually bookmarked this for myself for the next time I visit the island. It is close to the beach, under 7,500,000 Rupiah a month, away from tourists hotspots and has Internet access of some sort. Like most of the non-tourist areas, it is about 1-3 hours away from attractions but if you just want to sit and chill for a month, you'll be fine. If you do want to visit attractions, there are some available in the area but you'll probably be needing a driver, so add 500,000 rupiah to your budget for every day you want to go exploring.
Ahem, thats it. Sure, there may be more out there, I came up with a dozen hits but I weeded out the bad beaches (there are a lot of those) as well as the places that don't have Internet as well as the scam listings.
If you widened your search to include tourist areas, you'll increase your airbnb places to about 50 villas near beaches that you'd probably like. You'll also find that entertainment is cheaper as you won't have to spend bucketloads on private drivers and you'll have less of a culture shock if you haven't done SEA before.
You'll also find that hotels are cheaper than Airbnb, with some 3-4 star hotels willing to do rooms at around 250,000 Rupiah a night for a 4 week stay.
I bounce around Asia and find booking a place for a couple of days and searching for a place in person is much better than working with recommendations and Internet listings. Prices can be negotiated, neighbourhoods can be sussed out and seeing a place with your own eyes makes a huge difference.