Is it legal for an employer to ignore requests to verify employment?
I would say that if nothing else this would be a useful signal to you that your employment is not particularly secure. The employer may not want to expose themselves to liability to the landlord (or to you) by giving anyone an inaccurate level of confidence as to your security of income. This might be your cue to look for more modest (cheaper) accommodation, urgently reduce expenditures and debt generally, and start looking for other work.