Dorota - there are many good quality, established publishing houses that you could aim for. The main ones are Elsevier, Wiley-Blackwell, Palgrave-MacMillan, Sage, Oxford University Press, Cambridge University Press etc. However, not everyone 'enjoys' or wants to publish with large corporate publishing houses - especially on ethical or 'green' grounds. On the other hand, they can be more prestigious and are better equipped to 'internationalise' your book - and market it more widely.
Even though you have existing funds though Dorota - it's not as easy as 'just applying' to write a book. It's a competitive environment and you have to go through a robust proposal submission process that, not only takes time (which 5 months is not long - most texts take up to 2-years to get from proposal to publication) - but has to be agreed and accepted by the publishing editors.
Thank you very much, Dean and Joaquin. The problem is how to match prestige and quality with strict project time demands, so your advices are really helpful for me.