Polishing a Japanese sword is a major undertaking. There are only a handful of fully trained and competent togishi outside of Japan, and links to them can be found in the links section.
There are many others inside Japan, but please note that for any polisher, the waiting period can be from 6 months up to several years.
Within Japan, the laws for importing a sword make it easier to go through a broker. This is someone who will handle the import paperwork and deal directly with the polisher. This comes at a fee which can be a few hundred dollars, but is well worth it to be able to successfully complete the transaction with a minimum of hassle.
Outside of Japan, expect a similar waiting period. The better the polisher, the longer the waiting period usually.
Polishing comes at a significant price. The rate is usually between $50 and $150 per inch of blade cutting length. This means that an average katana of 25" blade length can cost on average $2500 for a decent polish. Then you have to understand that a newly polished blade usually requires a new habaki and shirasaya. Old scabbards usually have built up dirt and grime inside them that can scratch a newly polished blade, and therefore a new resting scabbard is recommended. The change in blade dimensions after a polish also require a newly fitted blade collar (habaki) afterwards. This can be as simple or as intricate as the customer desires. Expect a basic habaki to be in the region of $300 and a shirasaya to be around $400-600. These prices are estimates, and will vary depending on who is doing the work.