The Albright is a very good knot and I would be suspect of how it was tied. In Alaska 20lb to 40lb Salmon, even 60lb Kings, are caught all the time with out using a Bimini setup. Do you tie your Albright with a keeper on the tag end? You say your Albright knot broke. When a knot fails it is because it slips. I am not sure what you mean that the knot broke. Your failure could also be caused by using too light of a leader.
You can buy Bimini leaders that already have two Bimini's tied for you. You can get them in 16lb, 20lb and 22lb. They also make a Tarpon Bimini leader starting at a 60lb shock tippet. With a little practice you can tie your own Bimini. It is a lot easier than you think.
If you have an old piece of fly line it is great to practice with. Don't try to learn with a light piece of mono.
Rio Hand Tied Tarpon Leader
Item # GF009$8.95
RIO HAND-TIED "TARPON LEADERS" are the favorite among all of my buddies that guide for tarpon. Super tough 22-lb IGFA Class hard mono with 22-inches between the double biminis. 4-foot overall length. Complete leader and Huffnagle. Pre-straightened. SHOCK SIZES: 60-lb, 80-lb, 100-lb.
Here are some suggestions on how to fasten your backing to your fly line. The best method depends on how heavy of fly line you are using.
1. Using a single Nail Knot.
2. Using a double Nail Knot.
3. Using hollow braided mono with Nail Knots. This is a good methoid for heavier lines. Maybe the strongest method.
4. Using the Albright knot. Tie it with a keeper on the tag end.
5. Using two loops.
I use Pliobond on all of my permanate Nail Knots. Let me know if you need any help with any of these methoids. Here is the Guidebrod braided mono I am talking about in number three. It is not a leader like the label says. It is 50' of hollow braided mono.