Are there any passes that will help you avoid the lines at attractions in London?

The London Pass will help at some sites, not all. For it to make financial sense you need to plan carefully, but if money is not the issue....

In Paris, the Musuem Pass is phenomenal in value as well as useful in skipping the queue.

Which London attractions are you thinking of? Some you can buy tickets online in advance tomskipmthe ticket line ( Tower of London is one we did).

Discounts with Eurostar tickets? You must be thinking of the 241 offers for travel by train. The Eurostar does not count for those discounts. It does have 241 offers of its own for London and Paris museums:

http://www.eurostar.com/us-en/deals/2for1

However, most of the London museums listed are free entry! Maybe it works for the special exhibits that do have an entrance fee.

There are usually about 10 (5+5) offers to show your Eurostar ticket at your destination, either Paris or London, which show up in the magazine and perhaps, but not likely, on the booking website. They are not usually for prime attractions, usually a lower level of interest; so not much use for a short trip.

You can buy a membership to Historic Royal Palaces which will cover admission to the Tower of London, Hampton Court Palace, Kew Palace, Kensington Palace, the Banqueting House, and Hillsborough castle. It was worth it for us to buy because we had planned to visit 4 of the 6 places anyway. We bought the joint membership which covers 2 adults.
You'll still have to wait in short lines for security, but you won't have to wait in line for tickets.

I guess I would have to ask as well, what attractions are you looking at? In London it varies greatly by venue. There is a real mix of Trust properties, Private attractions, Paid Public, and free Public.

"I am not worried about the cost of the tickets. Just how to avoid the lines."

A lot of the attractions in London don't have any queues, if they do they usually move quickly, the exceptions being Westminster Abbey and the London Eye (but the queue is always moving). The main queues you may encounter are for security and you can't skip them.

The best way to avoid the lines is to buy your tickets online before you leave. Also try to go to the sights first thing in the morning before the crowds start getting heavy.