Charcoal, 60 bytes

Ｆ⊗Ｎ≔⁺⟦Ｅυ⊕λ⟧Ｅ∧υ⌊υ⁺ＬυＥ⮌υ§μλυＮθ≔⊟Φυ№ιθηＩΣＥΦυ‹↔⁻κ⌕υη²ΣΦι‹↔⁻μ⌕ηθ²

Try it online! Link is to verbose version of code. Explanation:

Ｆ⊗Ｎ≔⁺⟦Ｅυ⊕λ⟧Ｅ∧υ⌊υ⁺ＬυＥ⮌υ§μλυ

Input n and generate the spiral using the method from my answer to Print a NxN integer involute except 1-indexed.

Ｎθ≔⊟Φυ№ιθη

Input m and find the row containing m in the spiral.

ＩΣＥΦυ‹↔⁻κ⌕υη²ΣΦι‹↔⁻μ⌕ηθ²

Extract the neighbours of m and output the sum.

Charcoal, 60/62 bytes

Ｆ⊗Ｎ≔⁺⟦Ｅυ⊕λ⟧Ｅ∧υ⌊υ⁺ＬυＥ⮌υ§μλυＮθ≔⊟Φυ№ιθηＩΣＥΦυ‹↔⁻κ⌕υη²ΣΦι‹↔⁻μ⌕ηθ²

Try it online! Link is to verbose version of code. Explanation:

Ｆ⊗Ｎ≔⁺⟦Ｅυ⊕λ⟧Ｅ∧υ⌊υ⁺ＬυＥ⮌υ§μλυ

Input n and generate the spiral using the method from my answer to Print a NxN integer involute except 1-indexed.

Ｎθ≔⊟Φυ№ιθη

Input m and find the row containing m in the spiral.

ＩΣＥΦυ‹↔⁻κ⌕υη²ΣΦι‹↔⁻μ⌕ηθ²

Extract the neighbours of m and output the sum.

Note that the 5, 17 test case includes the 17 itself in the total; if this is not wanted then it can be subtracted at a cost of 2 bytes:

Ｆ⊗Ｎ≔⁺⟦Ｅυ⊕λ⟧Ｅ∧υ⌊υ⁺ＬυＥ⮌υ§μλυＮθ≔⊟Φυ№ιθηＩ⁻ΣＥΦυ‹↔⁻κ⌕υη²ΣΦι‹↔⁻μ⌕ηθ²θ

Try it online! Link is to verbose version of code.