How do you tell if they're representing you well? You see if the bills they said they'd introduce got introduced.

Not every Senator and Congressmen introduce bills during their term. Some even go as far to support a bill publically only to not vote for it when it comes to the floor. Now imagine that with a secret ballot. Now they could say they voted for a legislation even if they haven't because it makes them look good and it is literally against the law to find out whether what they say is true or not.

You see what they brought back for funding.

Most corporations can fund 501c (4) charities and pour money into a campaign without anyone knowing they've done it. So right now you have no clue where a majority of their funding is. During the 2012 elections there was over 500 million dollars unaccounted for. PAC's and Super PAC's are still the more outright, but if you make a secret ballot the corporations could simply funnel funding through other channels and now you wouldn't know who is voting for what and who is paying them.

You judge them on the job they're doing, like we do now.

What if they are investing in the future of the state or the country as a whole you would have no clue. Politicians currently want to keep their job and under a secret ballot they have every reason to lie to keep their job.

You can't both get rid of vote buying and keep open ballots.

Yes you can by not allowing hundreds of millions of dollars flowing into campaign funds. Lobbying is fueled by their ability to provide campaign funds. The thing you seem to forget is the politicians still need money for their campaigns which will only become harder in a secret ballot which means they have more incentive to get money from people that can afford it. Until you change the structure of how campaigns work, secret ballots don't fix anything. You can't assume that people who were corrupted before are just simply going to be less corrupt because they are invisible now, if anything the new anonymity gives them more incentive to lie more than they already do.

you could secretly be selling votes under the table.

Ok so a scenario under new campaign finance law. Both campaigns have about the same funds so their platform becomes more visible and stuff. Out of no where candidate B puts an extra million dollars in his campaign fund. Now we can immediately become suspicious of that, on top of the fact it is illegal. The campaign is the most important thing for politicians they fundraise for it incessantly, literally all year round. When you cut funding, and force their platforms forward we get a better election and then lobbyist have nothing to bring to the table when they introduce a bill to a legislator.