Proving that an $E$-oriented manifold has an $E$-oriented normal bundle

Question

This is the setting we are working in:

$M$ is a closed, smooth $n$-manifold embedded in $\mathbb{R}^{n+k}$ with a chosen embedding $e\colon M^n\to \mathbb{R}^{n+k}$. It is $E$-oriented, for $E$ a connected ring spectrum, i.e. there exists a fundamental class $$z \in E_n(M)$$ which is mapped, for every $m\in M$ to the generator of $E_n(M,M\setminus \{m\})$ via the inclusion $i_m\colon M \to M,M\setminus \{m\}$.

What I want to do: I want to follow the proof of Kochman (sadly the interesting page is not available for preview, it is Prop $4.3.5$ page 136-137 $b)$ implies $a)$.)

Little recap of proof: We will build local Thom classes for the restriction of the normal bundle and then proceed to glue them together using M-V. Let $\{N_i\}_{i=1}^s$ be a family of nbhds diffeo to open disks which over the manifold and let $z_i\in E_n(N_i,N_i\setminus m_i)$ be the image via excision and inclusion of the fundamental class.

Notice that we have non degenerate pairings $$E_n(N_i,N_i\setminus m_i)\otimes E^n(N_i,N_i\setminus m_i)\to \pi_0E$$ given suspension isomorphism and multiplication. These permit us to define the classes $z^i \in E^n(N_i,N_i\setminus m_i)$ as dual of the $z_i$'s. Then identifying $\nu^{-1}(N_i)/(\nu^{-1}(N_i)\setminus N_i)$ with the Thom space of the restriction of the normal bundle we have the following chain of identifications $$\widetilde{E}^k(\text{Th}(\nu_{|N_i}))\cong \widetilde{E}^k(D_n^+\wedge S^k)\cong \widetilde{E}^0(D^+_n)\cong \widetilde{E}^0(S^0)\cong \widetilde{E}^n(S^n)\cong E^n(N_i,N_i\setminus m_i) $$ which permits us to consider the class $t^i \in \widetilde{E}^k(\text{Th}(\nu_{|N_i}))$ define as the image through this chain (backwards) of the $z^i$'s.

Here is the problem: Kochman claims that after a standard application of M-V we can inductively build our global Thom class via a gluing of these $t^i$'s. What I can't prove is how to prove that these classes coincides on the intersection of the $N_i$'s. I tried writing down suitable diagrams but here are my thoughts:

1. We did a lot of non-canonical identifications, namely the pairing and the isos depend on some chosen homeomorphisms. This prevent us in finding a suitable commutative diagram in my opinion.
2. Compatibility at the level of orientation homological classes has to be checked going through $E_n(M)$, i.e. using the maps $$E_n(N_1,N_1\setminus m_1)\to E_n(M) \leftarrow E_n(N_2,N_2\setminus m_2)$$ There is no canonical way to go from $E_n(N_1,N_1\setminus m_1)\to E_n(N_2,N_2\setminus m_2)$. I can't translate this compatibility in a compatibility condition for the Thom classes
3. There is a similar argument on Switzer's book, page $319$ prop $14.18$. But there he uses micro-bundles and orientation as a Thom class of the tangent micro-bundle. I tried adapt the proof to no avail. (And it doesn't seem a trivial application of M-V).

So can someone give me some hints about this?

UPDATE: I've asked the question Math.Overflow and it seems from the comments that one has to use the so called Atyah-Duality. But this is a definitely non-trivial result. So either this represents a special case where the version of atyah duality one needs is very weak, or the proof doesn't work as it is