Heidelberg! A flat to rent wanted :)

There are two of us: me, a charming Product Manager in EdTech/Publishing (not deprived of confidence, but most importantly, solvent 😉) and my adorable cat. That one is not much of a breadwinner, but she sure knows how to steal hearts.
Starting from July, my career brings me to the Heidelberg branch of my company and for this reason, my cat and I are moving from Poland to Germany. Here's what we're on the hunt for:

- A cozy little haven, all to ourselves - no roommates, please!
- Ideally, 1.5 or 2 rooms, but hey, we're flexible if it's a steal.
- A kitchen that's ready to whip up some culinary magic.
- A trusty washing machine to keep our threads fresh and clean.
- Natural light, snug warmth and a building that isn't rundown.
- Good sanitary condition and no plumbing nightmares.
- Bonus points for a balcony, terrace or a tiny patch of greenery to call our own.
- Furniture is a plus, but it's not a deal-breaker.
- The flat doesn't have to be freshly renovated or boast a fancy design, I just want it to be a neutral and unimposing space with the potential for me to transform it into a cozy nest.
- Oh, and the landlord should to be cool with me hanging paintings and bookshelves on the walls.

Time frame? Long-term, from August.
The budget? 900 EUR month if I deal with the landlord directly. If there's an agent in the picture, much will depend on the amount of commission.