Update: See below for an attempt in LyX.
This is almost certainly not a suitable answer, but does demonstrate a possible way forward using babel, lualatex, and biblatex. babel is still pretty rough for RTL languages and biblatex doesn't support Hebrew either, so you would need to do some work to set strings up (but I think it could be done).
My code includes a couple of workarounds that I think shouldn't be necessary, but I had to make in order to get things working.
It works by specifying a langid field in your bibliography entries, then getting biblatex to change the language for every entry in the bibliography based on this (set up using autolang=other*).
\documentclass{article}
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@Book{מישהו1990,
title = {ספר},
publisher = {הוצאה},
date = {1990},
author = {מישהו},
langid = {hebrew}
}
@InCollection{sep-plato,
author = {Kraut, Richard},
title = {Plato},
booktitle = {The Stanford Encyclopedia of Philosophy},
publisher = {Metaphysics Research Lab, Stanford University},
date = {2017},
editor = {Edward N. Zalta},
edition = {Fall 2017},
url = {https://plato.stanford.edu/archives/fall2017/entries/plato/},
langid = {english}
}
\end{filecontents}
\usepackage[nil,bidi=basic-r,layout=lists]{babel}
\babelprovide[import=he,main]{hebrew}
\babelprovide[import=en]{english}
\babelfont{rm}[
Ligatures=TeX,
Extension=.ttf,
UprightFont=*-Medium,
ItalicFont=*-MediumOblique,
BoldFont=*-Bold,
BoldItalicFont=*-BoldOblique
]{TaameyDavidCLM}
\babelfont[american]{rm}[
Ligatures=TeX,
SmallCapsFont={Latin Modern Roman Caps}
]{Latin Modern Roman}
\babelfont{tt}[Ligatures=TeX]{Latin Modern Mono}
\usepackage[autolang=other*]{biblatex}
\addbibresource{\jobname.bib}
% set labels left to right, otherwise brackets are wrong. Why is this??
\DeclareFieldFormat{labelnumberwidth}{\textdir TLT \mkbibbrackets{#1}}
% slight tweak to bibliography environment
\defbibenvironment{bibliography}
{\list
{\printtext[labelnumberwidth]{%
\printfield{labelprefix}%
\printfield{labelnumber}}}
{\setlength{\labelwidth}{\labelnumberwidth}%
\setlength{\leftmargin}{\labelwidth}%
\setlength{\labelsep}{\biblabelsep}%
\addtolength{\leftmargin}{\labelsep}%
\setlength{\itemsep}{\bibitemsep}%
\setlength{\parsep}{\bibparsep}}%
\renewcommand*{\makelabel}[1]{\hss##1}}
{\endlist}
{\item \leavevmode}% add \leavevmode to work around babel quirk
\begin{document}
שלום \autocite{sep-plato}.
שלום \autocite{מישהו1990}.
\printbibliography[title=מקורות]
\end{document}

LyX File
You can actually do this in LyX. But it's a bit of a pain. And I'm not sure how robust it is. You have to do the language and biblatex setup in the preamble. Here's my attempt:
#LyX 2.3 created this file. For more info see http://www.lyx.org/
\lyxformat 544
\begin_document
\begin_header
\save_transient_properties true
\origin unavailable
\textclass article
\begin_preamble
\usepackage[nil,bidi=basic-r,layout=lists]{babel}
\babelprovide[import=he,main]{hebrew}
\babelprovide[import=en]{english}
\babelfont{rm}[
Ligatures=TeX,
Extension=.ttf,
UprightFont=*-Medium,
ItalicFont=*-MediumOblique,
BoldFont=*-Bold,
BoldItalicFont=*-BoldOblique
]{TaameyDavidCLM}
\babelfont[english]{rm}[
Ligatures=TeX,
SmallCapsFont={Latin Modern Roman Caps}
]{Latin Modern Roman}
\babelfont{tt}[Ligatures=TeX]{Latin Modern Mono}
\usepackage[style=numeric,autolang=other*]{biblatex}
% set labels left to right, otherwise brackets are wrong. Why is this??
\DeclareFieldFormat{labelnumberwidth}{\textdir TLT \mkbibbrackets{#1}}
% slight tweak to bibliography environment
\defbibenvironment{bibliography}
{\list
{\printtext[labelnumberwidth]{%
\printfield{labelprefix}%
\printfield{labelnumber}}}
{\setlength{\labelwidth}{\labelnumberwidth}%
\setlength{\leftmargin}{\labelwidth}%
\setlength{\labelsep}{\biblabelsep}%
\addtolength{\leftmargin}{\labelsep}%
\setlength{\itemsep}{\bibitemsep}%
\setlength{\parsep}{\bibparsep}}%
\renewcommand*{\makelabel}[1]{\hss##1}}
{\endlist}
{\item \leavevmode}% add \leavevmode to work around babel quirk
% set references string, since biblatex falls back to english
\DefineBibliographyStrings{english}{
references = {מקורות}
}
\end_preamble
\use_default_options true
\maintain_unincluded_children false
\language english
\language_package none
\inputencoding default
\fontencoding global
\font_roman "default" "default"
\font_sans "default" "default"
\font_typewriter "default" "default"
\font_math "auto" "auto"
\font_default_family default
\use_non_tex_fonts true
\font_sc false
\font_osf false
\font_sf_scale 100 100
\font_tt_scale 100 100
\use_microtype false
\use_dash_ligatures true
\graphics default
\default_output_format pdf5
\output_sync 0
\bibtex_command biber
\index_command default
\paperfontsize default
\spacing single
\use_hyperref false
\papersize default
\use_geometry false
\use_package amsmath 1
\use_package amssymb 1
\use_package cancel 1
\use_package esint 1
\use_package mathdots 1
\use_package mathtools 1
\use_package mhchem 1
\use_package stackrel 1
\use_package stmaryrd 1
\use_package undertilde 1
\cite_engine biblatex
\cite_engine_type numerical
\biblio_style plain
\biblio_options autolang=other*
\biblatex_bibstyle numeric
\biblatex_citestyle numeric
\use_bibtopic false
\use_indices false
\paperorientation portrait
\suppress_date false
\justification true
\use_refstyle 1
\use_minted 0
\index Index
\shortcut idx
\color #008000
\end_index
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\paragraph_indentation default
\is_math_indent 0
\math_numbering_side default
\quotes_style english
\dynamic_quotes 0
\papercolumns 1
\papersides 1
\paperpagestyle default
\tracking_changes false
\output_changes false
\html_math_output 0
\html_css_as_file 0
\html_be_strict false
\end_header
\begin_body
\begin_layout Standard
\lang hebrew
שלום
\begin_inset CommandInset citation
LatexCommand autocite
key "sep-plato"
literal "false"
\end_inset
.
\end_layout
\begin_layout Standard
\lang hebrew
שלום
\begin_inset CommandInset citation
LatexCommand autocite
key "מישהו1990"
literal "false"
\end_inset
.
\end_layout
\begin_layout Standard
This is in English.
\end_layout
\begin_layout Standard
\lang hebrew
\begin_inset CommandInset bibtex
LatexCommand bibtex
btprint "btPrintCited"
bibfiles "simplebib"
\end_inset
\end_layout
\end_body
\end_document
And simplebib.bib:
@Book{מישהו1990,
title = {ספר},
publisher = {הוצאה},
date = {1990},
author = {מישהו},
langid = {hebrew}
}
@InCollection{sep-plato,
author = {Kraut, Richard},
title = {Plato},
booktitle = {The Stanford Encyclopedia of Philosophy},
publisher = {Metaphysics Research Lab, Stanford University},
date = {2017},
editor = {Edward N. Zalta},
edition = {Fall 2017},
url = {https://plato.stanford.edu/archives/fall2017/entries/plato/},
langid = {english}
}
Output is basically the same as above. (Except with the English paragraph added as well.)
simplebib.bibfile. (Your screenshot suggests at least the Hebrew entry is different.) – David Purton Aug 27 '18 at 15:24babel(currently 3.22) – David Purton Aug 27 '18 at 15:29xelatex. It is only forlualatex. I think when I originally looked at your question. I tried withxelatex, but couldn't do it. – David Purton Aug 27 '18 at 16:25TaameyDavidCLMit says that the font doesn't found and I can't write at Sans-Serif fonts.... – heblyx Aug 27 '18 at 16:30