I think that the best way is:
1. Protect the amino group of the amino acid leucine with the Fmoc group
2. Join the carbonyl end of the amino acid Leu to a reactive group of a resin.
3. Remove the protective group by treatment with a solution containing a weak organic base.
4. Protect the valine amino group with Fmoc. The amino acid is activated in its carboxyl group by DCC.
5. The amino group of leucine attacks the activated carboxyl group of valine to form a peptide bond.
6. We unprotect the Leu-Val di-peptide as in step 2. We add HF to hydrolyze the ester bond between the resin and the di-peptide.
7. Once we have the Leu-Val di-peptide, we protect the amino end with the Fmoc group, and proceed as before.
You are wasting time, you can buy fmoc-AAs commercialy for cheap money. While your idea might work its way too inefficient for this job:
a) you dont have to cleave the peptide from the resin only to attach it again
b) you will get even lower yields than normally
c) working with HF is not needed.
In SPS usually, the aminoacids are immobilized as esters to the resin and the you deprotect-acylate and repeate untill you get the peptide. You can start by immobilizing at N-terminus or even by side chain but its far less common.
Here is what I would do:
1) connect the fmoc-phenylalanine to wang resin using some set of obvious conditions (look it up, I think its DIC+dmap but im not 100%).
2) analyze the loading of the resin via cleaving the fmoc-AA from the resin and doing quantification
3) de-Fmoc with (20%) piperidine in DCM, should be super fast
4) make the peptide bond using another set of obvious conditions (DIC+HOBt or any other you wish)
5) repeat 3+4
6) cleave the peptide from resin using TFA:DCM cocktail and purify by HPLC