Uiua

[Works on test data, but fails on Part 3 live data: my checksum is "Correct length and correct first character". Posting for the record while I think about it more.] Turned out to be trailing zeros.

F ← (
  ↘1⊙[0_0_0] # Create empty row
  ∧(
    ◡≡([⊃(↧⊃(=0⊢◌|>⊙⊡₁)|=0⊡₁◌|↧⊃(=0⊣◌|<⊙⊡₁))]) # Will it fit in each row?
    ⊟⊙(⊢⊚)⟜⊡⊢⊚⊸≡/↥                             # Find first non-zero row and col.
    (⍜⊡⋅∘)⊙⊃⋅∘∘                                # insert
    ⍥(˜⊂0_0_0)¬≍0_0_0⊸⊣                        # Add padding row back
  )
  ↘¯1
)
S     ← ⋕/$"__"⊡1⍉
Part₁ ← S F
Part₂ ← -⊃/↧/↥≡(S F)

Part₃ ← (
  ⬚0⊸≡(⊂⊃(S|≡(°⊥₁₀▽⊸>₀⇌))F) # All sword stats
  ≡⊂⊙(⊢⍉)             # Append sword IDs
  ⊏⊸⍖                 # Sort
  /+×+1°⊏≡⊣           # Checksum
)

Part₁ [58 5 3 7 8 9 10 4 5 7 8 8]
Part₂ [[2 7 9 9 3 8 3 8 8 6 8] [5 2 9 3 8 3 9 5 2 1 4]]
Part₃ [[1 7 1 9 1 6 9 8 3 7 2]
       [2 7 1 9 1 6 9 8 3 7 2]]