Three glorious old continetal style cafes worth visiting for cakes and coffee are:
Ruswurm (Buda Side - tucked down street near castle
Gerbaud (Buda side - very polular)
Cafe inside Hotel Astoria
All three are atmospheric, great coffee and cakes and marzipan stuff for those with a sweet tooth
www.ruszwurm.hu
www.gerbeaud.hu
www.danubiushotels.com/en/budapest-hotels/hotel-astoria-budapest
Updates
Gerbaud is on the Pest side (Vörösmarty tér) city centre end-of-the-line on Metro 1 Gerbauld's lived up to all my expectations of Hapsburgian coffeehouses, truly Belle Epoque! Great coffee, attentive service, fabulous location for watching the world go by and thinking about finally writing that novel.
Is this your business?
Another fantastic place to visit is the "Daubner Cukrászda", though it's only take-away. Be prepared to queue up for 15-30 minutes, but their cakes and ice cream are worth the wait!
To get to Daubner take the HÉV from Batthyány tér to Szépvölgyi út and walk up the hill for about 5 mins. it's a one-storey building on the left.