Let's build a Staircase

Python 3, 264 Bytes

I'm not good at code golf so I'm confident this won't be the best Python 3 answer. This uses recursion and an ordered dict with all the words for each length.

from collections import*
def f(i):
 d = defaultdict(list)
 for l in i: x = "".join(c for c in l if c.isalnum());d[len(x)].append(x)
 n = (sorted(["".join(d[z]) for z in d.keys()], key=len))
 if n == i:return "\n".join(n)
 return f(n)
print(f(eval(input())))

Takes input from stdin as a list, for example, test it with this list:

['A', 'small', 'one', 'that', 'contains', 'equal', 'length', 'strings', 'for', 'the', 'special', 'rule']

Will output:

A
length
oneforthe
smallequal
stringsspecial
thatrulecontains

Retina, 69 63 bytes

[^\w¶]|_

{`\b((.)+)¶((?.)+)\b(?(2)(?!))
$1$3
O$`(.)+
$#1$*

Try it online!

Husk, 11 bytes

ωȯmΣġLÖLmf□

Try it online!

Husk is younger than this challenge (which makes no difference officially, but still).

Explanation

ωȯmΣġLÖLmf□  Implicit input (list of strings), say ["a","bc","d!","123"]
        mf□  Keep only alphanumeric chars of each: ["a","bc","d","123"]
ωȯ           Repeat until fixed point is reached:
      ÖL       Sort by length: ["a","d","bc","123"]
    ġL         Group by length: [["a","d"],["bc"],["123"]]
  mΣ           Concatenate each group: ["ad","bc","123"]
             Final result ["123","adbc"], print implicitly separated by newlines.