I think the trip is doable reasonable well, although double or triple the time wouldn't be a bad thing. You just have to concentrate on the main sights. For example in Paris, you could easily spend one or two days in the Louvre, but if you're happy with seeing it only from outside, or just take a glimpse at Mona Lisa, 1 hours is enough.
I traveled really a lot in Europe by car, train, plane, etc and my recommendation would be the following:
Day 1: Arrive early in the morning in London and spend the rest of the day sightseeing.
Day 2: Spend the whole day in London sightseeing
Day 3: Early in the morning leave London by train and go to Brussels. You can be before 10:00 in the center of Brussels. Then do some sightseeing there.
Day 4: Do sightseeing in Brussels.
Day 5: Leave early in the morning for Amsterdam. The trip takes only 2.5 hours, so you will be in Amsterdam easily before noon. The rest of the day you can spend in Amsterdam.
Day 6: Do some sightseeing in Amsterdam. In the evening take a night train to Munich and you will be in Munich before 8! In the night train you can sleep reasonable well and so you will be fit for the whole day sightseeing in Munich. I would recommend to visit the Olympic Stadium, The Hofbräuhaus, The City Center (e.g. Marienplatz, Frauenkirche, Neues Rathaus, etc.) and Thalkrichen to relax a little bit. If you're a big soccer fan you should watch a game of Bayern Munich in the evening in the beautiful Allianz Arena.
Day 7: Still sightseeing in Munich. Do the things you missed the day before ;)
Day 8: Head to Switzerland. Here it really depends on what you're interested in. If you're interested in cities, I would go to Bern, Luzern, Basel or Zürich. IF you're more a nature fan and if you want to see what Switzerland is really famous for, I would spend some time in more remote areas, namely in the high Alps ;) There I could recommend to visit small villages in Engadin or Graubünden. But the Berner Oberland is nice too. Really worth a trip is the Top of Europe! You can take a train up to 3400 meters above see level and then walk there on a glacier. You will have a spectacular view to see some of the highest mountains in Europe. For the further recommendations I assume that on day 8 you're headed to Berne to visit the city in one day. The train connection from Munich to Berne isn't really very good (takes more than 6 hours). So I would take a look if someone can take you with you in his car. There are some big internet portals that offer so called Mitfahrgelegenheit. Normally it is no problem to find someone that takes you for a really cheap price from Munich to somewhere in Switzerland. Once you're in Switzerland you can use the excellent railway system to rush to Berne.
Day 9: Leave Berne and make a day trip to the Top of Europe as mentioned before. In the evening take a night train to Rome (and skip Pisa ;) it's just that weird tower, nothing special ;)). Almost all night trains from Berne go via Zurich and Milano. You can be in Rome at sunrise.
Day 10: Arrive early morning in Rome and spend some time sightseeing. I can recommend the following sights: Colosseo, the Vatican, the Pantheon, Roman Forum, Pyramid of Cestus, Castel Sant'Angelo (well known the Dan Brown books), Capitoline Hill, the world famous Spanish Steps, the also very famous Trevi Fountain and the bridge Ponte Sant'Angelo that least to the castle with the same name.
Day 11: Still sightseeing in Rome.
Day 12: Take an early budget flight to Paris and spend the rest of the day there.
Day 13: Sightseeing in Paris. I can recommend the following sights: Arc de Triomphe, Château de Versailles ( a little bit outside the city, so only if you have enough time), The Eiffel Tower (Tour Eiffel), La Défense, Notre Dame, Pantheon, Père-Lachaise Cementery, Sacré Coeur, The Louvre, Musée d'Orsay, Centre Georges Pompidou and Les Invalides.
Day 14: Sightseeing in Paris and then take the Eurostar to London. This will only take 2:15!
So you see it is doable, but you have to spend a lot of time in trains, and you will have to take night trains. Another problem is that the price level in Switzerland is really high. So your 2000 USD are not too many in Switzerland.